Mobile guided surface waveguide probes and receivers

ABSTRACT

The present disclosure is directed to mobile guided surface waveguide probes and receivers. In a representative embodiment, an excitation source such as a generator is coupled to a guided surface waveguide probe. The excitation source and the guided surface waveguide probe are mounted to a rigid frame for transport.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. application Ser. No. 15/238,960, filed Aug. 17, 2016, entitled “Mobile Guided Surface Waveguide Probes and Receivers”, claims priority to, and the benefit of, U.S. Provisional Patent Application entitled “Mobile Guided Surface Waveguide Probes and Receivers,” which was filed on Sep. 10, 2015 and assigned Application No. 62/216,747. Both applications, are incorporated herein by reference in their entireties. This application is related to co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7, 2013 and assigned application Ser. No. 13/789,538, and was published on Sep. 11, 2014 as Publication Number US2014/0252886 A1, and which is incorporated herein by reference in its entirety. This application is also related to co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7, 2013 and assigned application Ser. No. 13/789,525, and was published on Sep. 11, 2014 as Publication Number US2014/0252865 A1, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on Sep. 10, 2014 and assigned application Ser. No. 14/483,089, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Waves,” which was filed on Jun. 2, 2015 and assigned Application Ser. No. 14/728,507, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Waves,” which was filed on Jun. 2, 2015 and assigned application Ser. No. 14/728,492, and which is incorporated herein by reference in its entirety.

BACKGROUND

For over a century, signals transmitted by radio waves involved radiation fields launched using conventional antenna structures. In contrast to radio science, electrical power distribution systems in the last century involved the transmission of energy guided along electrical conductors. This understanding of the distinction between radio frequency (RF) and power transmission has existed since the early 1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views.

FIG. 1 is a chart that depicts field strength as a function of distance for a guided electromagnetic field and a radiated electromagnetic field.

FIG. 2 is a drawing that illustrates a propagation interface with two regions employed for transmission of a guided surface wave according to various embodiments of the present disclosure.

FIG. 3 is a drawing that illustrates a guided surface waveguide probe disposed with respect to a propagation interface of FIG. 2 according to various embodiments of the present disclosure.

FIG. 4 is a plot of an example of the magnitudes of close-in and far-out asymptotes of first order Hankel functions according to various embodiments of the present disclosure.

FIGS. 5A and 5B are drawings that illustrate a complex angle of incidence of an electric field synthesized by a guided surface waveguide probe according to various embodiments of the present disclosure.

FIG. 6 is a graphical representation illustrating the effect of elevation of a charge terminal on the location where the electric field of FIG. 5A intersects with the lossy conducting medium at a Brewster angle according to various embodiments of the present disclosure.

FIG. 7 is a graphical representation of an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

FIGS. 8A through 8C are graphical representations illustrating examples of equivalent image plane models of the guided surface waveguide probe of FIGS. 3 and 7 according to various embodiments of the present disclosure.

FIGS. 9A and 9B are graphical representations illustrating examples of single-wire transmission line and classic transmission line models of the equivalent image plane models of FIGS. 8B and 8C according to various embodiments of the present disclosure.

FIG. 10 is a flow chart illustrating an example of adjusting a guided surface waveguide probe of FIGS. 3 and 7 to launch a guided surface wave along the surface of a lossy conducting medium according to various embodiments of the present disclosure.

FIG. 11 is a plot illustrating an example of the relationship between a wave tilt angle and the phase delay of a guided surface waveguide probe of FIGS. 3 and 7 according to various embodiments of the present disclosure.

FIG. 12 is a drawing that illustrates an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

FIG. 13 is a graphical representation illustrating the incidence of a synthesized electric field at a complex Brewster angle to match the guided surface waveguide mode at the Hankel crossover distance according to various embodiments of the present disclosure.

FIG. 14 is a graphical representation of an example of a guided surface waveguide probe of FIG. 12 according to various embodiments of the present disclosure.

FIG. 15A includes plots of an example of the imaginary and real parts of a phase delay (Φ_(U)) of a charge terminal T₁ of a guided surface waveguide probe according to various embodiments of the present disclosure.

FIG. 15B is a schematic diagram of the guided surface waveguide probe of FIG. 14 according to various embodiments of the present disclosure.

FIG. 16 is a drawing that illustrates an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

FIG. 17 is a graphical representation of an example of a guided surface waveguide probe of FIG. 16 according to various embodiments of the present disclosure.

FIGS. 18A through 18C depict examples of receiving structures that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

FIG. 18D is a flow chart illustrating an example of adjusting a receiving structure according to various embodiments of the present disclosure.

FIG. 19 depicts an example of an additional receiving structure that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

FIGS. 20A through 20E are schematic symbols used to represent the guided surface waveguide probe and receiving structures according to the various embodiments of the present disclosure.

FIG. 21 is an illustration depicting an example of employing mobile power transmitters, mobile power repeaters and power reception kits according to the various embodiments of the present disclosure.

FIG. 22A is a diagram that illustrates an example of a mobile power transmitter according to the various embodiments of the present disclosure.

FIG. 22B is a diagram that illustrates an example of a mobile power repeater according to the various embodiments of the present disclosure.

FIG. 23 is an illustration depicting an example of a mobile power transmitter according to the various embodiments of the present disclosure.

FIGS. 24A and 24B are illustrations depicting an example of a power reception kit according to the various embodiments of the present disclosure.

FIGS. 25A and 25B are illustrations depicting another example of a power reception kit according to the various embodiments of the present disclosure.

FIGS. 26A and 26B are illustrations depicting a further example of a power reception kit according to the various embodiments of the present disclosure.

FIGS. 27A and 27B are illustrations depicting a further example of a power reception kit according to the various embodiments of the present disclosure.

FIGS. 28A and 28B are illustrations depicting examples of interconnected power reception kits according to the various embodiments of the present disclosure.

FIG. 29 is a diagram that illustrates an example of the power output circuitry of a power reception kit according to the various embodiments of the present disclosure.

FIG. 30 is a flow chart illustrating an example of the operation of a power reception kit.

DETAILED DESCRIPTION

To begin, some terminology shall be established to provide clarity in the discussion of concepts to follow. First, as contemplated herein, a formal distinction is drawn between radiated electromagnetic fields and guided electromagnetic fields.

As contemplated herein, a radiated electromagnetic field comprises electromagnetic energy that is emitted from a source structure in the form of waves that are not bound to a waveguide. For example, a radiated electromagnetic field is generally a field that leaves an electric structure such as an antenna and propagates through the atmosphere or other medium and is not bound to any waveguide structure. Once radiated electromagnetic waves leave an electric structure such as an antenna, they continue to propagate in the medium of propagation (such as air) independent of their source until they dissipate regardless of whether the source continues to operate. Once electromagnetic waves are radiated, they are not recoverable unless intercepted, and, if not intercepted, the energy inherent in the radiated electromagnetic waves is lost forever. Electrical structures such as antennas are designed to radiate electromagnetic fields by maximizing the ratio of the radiation resistance to the structure loss resistance. Radiated energy spreads out in space and is lost regardless of whether a receiver is present. The energy density of the radiated fields is a function of distance due to geometric spreading. Accordingly, the term “radiate” in all its forms as used herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wave whose energy is concentrated within or near boundaries between media having different electromagnetic properties. In this sense, a guided electromagnetic field is one that is bound to a waveguide and may be characterized as being conveyed by the current flowing in the waveguide. If there is no load to receive and/or dissipate the energy conveyed in a guided electromagnetic wave, then no energy is lost except for that dissipated in the conductivity of the guiding medium. Stated another way, if there is no load for a guided electromagnetic wave, then no energy is consumed. Thus, a generator or other source generating a guided electromagnetic field does not deliver real power unless a resistive load is present. To this end, such a generator or other source essentially runs idle until a load is presented. This is akin to running a generator to generate a 60 Hertz electromagnetic wave that is transmitted over power lines where there is no electrical load. It should be noted that a guided electromagnetic field or wave is the equivalent to what is termed a “transmission line mode.” This contrasts with radiated electromagnetic waves in which real power is supplied at all times in order to generate radiated waves. Unlike radiated electromagnetic waves, guided electromagnetic energy does not continue to propagate along a finite length waveguide after the energy source is turned off. Accordingly, the term “guide” in all its forms as used herein refers to this transmission mode of electromagnetic propagation.

Referring now to FIG. 1, shown is a graph 100 of field strength in decibels (dB) above an arbitrary reference in volts per meter as a function of distance in kilometers on a log-dB plot to further illustrate the distinction between radiated and guided electromagnetic fields. The graph 100 of FIG. 1 depicts a guided field strength curve 103 that shows the field strength of a guided electromagnetic field as a function of distance. This guided field strength curve 103 is essentially the same as a transmission line mode. Also, the graph 100 of FIG. 1 depicts a radiated field strength curve 106 that shows the field strength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103 and 106 for guided wave and for radiation propagation, respectively. The radiated field strength curve 106 falls off geometrically (1/d, where d is distance), which is depicted as a straight line on the log-log scale. The guided field strength curve 103, on the other hand, has a characteristic exponential decay of e^(−αd)/√{square root over (d)} and exhibits a distinctive knee 109 on the log-log scale. The guided field strength curve 103 and the radiated field strength curve 106 intersect at point 112, which occurs at a crossing distance. At distances less than the crossing distance at intersection point 112, the field strength of a guided electromagnetic field is significantly greater at most locations than the field strength of a radiated electromagnetic field. At distances greater than the crossing distance, the opposite is true. Thus, the guided and radiated field strength curves 103 and 106 further illustrate the fundamental propagation difference between guided and radiated electromagnetic fields. For an informal discussion of the difference between guided and radiated electromagnetic fields, reference is made to Milligan, T., Modern Antenna Design, McGraw-Hill, 1^(st) Edition, 1985, pp. 8-9, which is incorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, made above, is readily expressed formally and placed on a rigorous basis. That two such diverse solutions could emerge from one and the same linear partial differential equation, the wave equation, analytically follows from the boundary conditions imposed on the problem. The Green function for the wave equation, itself, contains the distinction between the nature of radiation and guided waves.

In empty space, the wave equation is a differential operator whose eigenfunctions possess a continuous spectrum of eigenvalues on the complex wave-number plane. This transverse electro-magnetic (TEM) field is called the radiation field, and those propagating fields are called “Hertzian waves.” However, in the presence of a conducting boundary, the wave equation plus boundary conditions mathematically lead to a spectral representation of wave-numbers composed of a continuous spectrum plus a sum of discrete spectra. To this end, reference is made to Sommerfeld, A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,” Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A., “Problems of Radio,” published as Chapter 6 in Partial Differential Equations in Physics—Lectures on Theoretical Physics: Volume VI, Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “Hertzian Dipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20th Century Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46, No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss, H. L., and Skalnik, J. G., Microwave Theory and Techniques, Van Nostrand, 1953, pp. 291-293, each of these references being incorporated herein by reference in its entirety.

The terms “ground wave” and “surface wave” identify two distinctly different physical propagation phenomena. A surface wave arises analytically from a distinct pole yielding a discrete component in the plane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves” by Cullen, A. L., (Proceedings of the IEE (British), Vol. 101, Part IV, August 1954, pp. 225-235). In this context, a surface wave is considered to be a guided surface wave. The surface wave (in the Zenneck-Sommerfeld guided wave sense) is, physically and mathematically, not the same as the ground wave (in the Weyl-Norton-FCC sense) that is now so familiar from radio broadcasting. These two propagation mechanisms arise from the excitation of different types of eigenvalue spectra (continuum or discrete) on the complex plane. The field strength of the guided surface wave decays exponentially with distance as illustrated by curve 103 of FIG. 1 (much like propagation in a lossy waveguide) and resembles propagation in a radial transmission line, as opposed to the classical Hertzian radiation of the ground wave, which propagates spherically, possesses a continuum of eigenvalues, falls off geometrically as illustrated by curve 106 of FIG. 1, and results from branch-cut integrals. As experimentally demonstrated by C. R. Burrows in “The Surface Wave in Radio Propagation over Plane Earth” (Proceedings of the IRE, Vol. 25, No. 2, February, 1937, pp. 219-229) and “The Surface Wave in Radio Transmission” (Bell Laboratories Record, Vol. 15, June 1937, pp. 321-324), vertical antennas radiate ground waves but do not launch guided surface waves.

To summarize the above, first, the continuous part of the wave-number eigenvalue spectrum, corresponding to branch-cut integrals, produces the radiation field, and second, the discrete spectra, and corresponding residue sum arising from the poles enclosed by the contour of integration, result in non-TEM traveling surface waves that are exponentially damped in the direction transverse to the propagation. Such surface waves are guided transmission line modes. For further explanation, reference is made to Friedman, B., Principles and Techniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286, 290, 298-300.

In free space, antennas excite the continuum eigenvalues of the wave equation, which is a radiation field, where the outwardly propagating RF energy with E_(z) and H_(ϕ) in-phase is lost forever. On the other hand, waveguide probes excite discrete eigenvalues, which results in transmission line propagation. See Collin, R. E., Field Theory of Guided Waves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoretical analyses have held out the hypothetical possibility of launching open surface guided waves over planar or spherical surfaces of lossy, homogeneous media, for more than a century no known structures in the engineering arts have existed for accomplishing this with any practical efficiency. Unfortunately, since it emerged in the early 1900's, the theoretical analysis set forth above has essentially remained a theory and there have been no known structures for practically accomplishing the launching of open surface guided waves over planar or spherical surfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, various guided surface waveguide probes are described that are configured to excite electric fields that couple into a guided surface waveguide mode along the surface of a lossy conducting medium. Such guided electromagnetic fields are substantially mode-matched in magnitude and phase to a guided surface wave mode on the surface of the lossy conducting medium. Such a guided surface wave mode can also be termed a Zenneck waveguide mode. By virtue of the fact that the resultant fields excited by the guided surface waveguide probes described herein are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided electromagnetic field in the form of a guided surface wave is launched along the surface of the lossy conducting medium. According to one embodiment, the lossy conducting medium comprises a terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides for an examination of the boundary value solutions to Maxwell's equations derived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck, J., “On the Propagation of Plane Electromagnetic Waves Along a Flat Conducting Surface and their Relation to Wireless Telegraphy,” Annalen der Physik, Serial 4, Vol. 23, Sep. 20, 1907, pp. 846-866. FIG. 2 depicts cylindrical coordinates for radially propagating waves along the interface between a lossy conducting medium specified as Region 1 and an insulator specified as Region 2. Region 1 can comprise, for example, any lossy conducting medium. In one example, such a lossy conducting medium can comprise a terrestrial medium such as the Earth or other medium. Region 2 is a second medium that shares a boundary interface with Region 1 and has different constitutive parameters relative to Region 1. Region 2 can comprise, for example, any insulator such as the atmosphere or other medium. The reflection coefficient for such a boundary interface goes to zero only for incidence at a complex Brewster angle. See Stratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the present disclosure sets forth various guided surface waveguide probes that generate electromagnetic fields that are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium comprising Region 1. According to various embodiments, such electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium that can result in zero reflection.

To explain further, in Region 2, where an e^(jωt) field variation is assumed and where ρ≠0 and z≥0 (with z being the vertical coordinate normal to the surface of Region 1, and ρ being the radial dimension in cylindrical coordinates), Zenneck's closed-form exact solution of Maxwell's equations satisfying the boundary conditions along the interface are expressed by the following electric field and magnetic field components:

$\begin{matrix} {{H_{2\varphi} = {{Ae}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},} & (1) \\ {{E_{2\; \rho} = {{A\left( \frac{u_{2}}{j\; {\omega ɛ}_{0}} \right)}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},{and}} & (2) \\ {E_{2\; z} = {{A\left( \frac{- \gamma}{{\omega ɛ}_{0}} \right)}e^{{- u_{2}}z}H_{0}^{(2)}{\left( {{- j}\; {\gamma\rho}} \right).}}} & (3) \end{matrix}$

In Region 1, where the e^(jωt) field variation is assumed and where ρ≠0 and z≤0, Zenneck's closed-form exact solution of Maxwell's equations satisfying the boundary conditions along the interface is expressed by the following electric field and magnetic field components:

$\begin{matrix} {{H_{1\varphi} = {{Ae}^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},} & (4) \\ {{E_{1\; \rho} = {{A\left( \frac{- u_{1}}{\sigma_{1} + {j\; {\omega ɛ}_{1}}} \right)}e^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},{and}} & (5) \\ {E_{1\; z} = {{A\left( \frac{{- j}\; \gamma}{\sigma_{1} + {j\; {\omega ɛ}_{1}}} \right)}e^{u_{1}z}H_{0}^{(2)}{\left( {{- j}\; {\gamma\rho}} \right).}}} & (6) \end{matrix}$

In these expressions, z is the vertical coordinate normal to the surface of Region 1 and ρ is the radial coordinate, H_(n) ⁽²⁾(−jγρ) is a complex argument Hankel function of the second kind and order n, u₁ is the propagation constant in the positive vertical (z) direction in Region 1, u₂ is the propagation constant in the vertical (z) direction in Region 2, σ₁ is the conductivity of Region 1, ω is equal to 2πf, where f is a frequency of excitation, ε_(o) is the permittivity of free space, ε₁ is the permittivity of Region 1, A is a source constant imposed by the source, and γ is a surface wave radial propagation constant.

The propagation constants in the ±z directions are determined by separating the wave equation above and below the interface between Regions 1 and 2, and imposing the boundary conditions. This exercise gives, in Region 2,

$\begin{matrix} {u_{2} = \frac{- {jk}_{0}}{\sqrt{1 + \left( {ɛ_{r} + {jx}} \right)}}} & (7) \end{matrix}$

and gives, in Region 1,

u ₁ =−u ₂(ε_(r) −jx).  (8)

The radial propagation constant γ is given by

$\begin{matrix} {{\gamma = {{j\sqrt{k_{0}^{2} + u_{2}^{2}}} = {j\frac{k_{0}n}{\sqrt{1 + n^{2}}}}}},} & (9) \end{matrix}$

which is a complex expression where n is the complex index of refraction given by

n=√{square root over (ε_(r) −jx)}.  (10)

In all of the above Equations,

$\begin{matrix} {{x = \frac{\sigma_{1}}{{\omega ɛ}_{0}}},\mspace{14mu} {and}} & (11) \\ {{k_{0} = {{\omega \sqrt{\mu_{0}ɛ_{0}}} = \frac{\lambda_{0}}{2\pi}}},} & (12) \end{matrix}$

where ε_(r) comprises the relative permittivity of Region 1, σ₁ is the conductivity of Region 1, ε_(o) is the permittivity of free space, and μ_(o) comprises the permeability of free space. Thus, the generated surface wave propagates parallel to the interface and exponentially decays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considered to be a cylindrically-symmetric, radially-propagating waveguide mode. See Barlow, H. M., and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 10-12, 29-33. The present disclosure details structures that excite this “open boundary” waveguide mode. Specifically, according to various embodiments, a guided surface waveguide probe is provided with a charge terminal of appropriate size that is fed with voltage and/or current and is positioned relative to the boundary interface between Region 2 and Region 1. This may be better understood with reference to FIG. 3, which shows an example of a guided surface waveguide probe 200 a that includes a charge terminal T₁ elevated above a lossy conducting medium 203 (e.g., the Earth) along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. The lossy conducting medium 203 makes up Region 1, and a second medium 206 makes up Region 2 and shares a boundary interface with the lossy conducting medium 203.

According to one embodiment, the lossy conducting medium 203 can comprise a terrestrial medium such as the planet Earth. To this end, such a terrestrial medium comprises all structures or formations included thereon whether natural or man-made. For example, such a terrestrial medium can comprise natural elements such as rock, soil, sand, fresh water, sea water, trees, vegetation, and all other natural elements that make up our planet. In addition, such a terrestrial medium can comprise man-made elements such as concrete, asphalt, building materials, and other man-made materials. In other embodiments, the lossy conducting medium 203 can comprise some medium other than the Earth, whether naturally occurring or man-made. In other embodiments, the lossy conducting medium 203 can comprise other media such as man-made surfaces and structures such as automobiles, aircraft, man-made materials (such as plywood, plastic sheeting, or other materials) or other media.

In the case where the lossy conducting medium 203 comprises a terrestrial medium or Earth, the second medium 206 can comprise the atmosphere above the ground. As such, the atmosphere can be termed an “atmospheric medium” that comprises air and other elements that make up the atmosphere of the Earth. In addition, it is possible that the second medium 206 can comprise other media relative to the lossy conducting medium 203.

The guided surface waveguide probe 200 a includes a feed network 209 that couples an excitation source 212 to the charge terminal T₁ via, e.g., a vertical feed line conductor. According to various embodiments, a charge Q₁ is imposed on the charge terminal T₁ to synthesize an electric field based upon the voltage applied to terminal T₁ at any given instant. Depending on the angle of incidence (θ_(i)) of the electric field (E), it is possible to substantially mode-match the electric field to a guided surface waveguide mode on the surface of the lossy conducting medium 203 comprising Region 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6), the Leontovich impedance boundary condition between Region 1 and Region 2 can be stated as

{circumflex over (z)}×

₂(ρ,φ,0)=

_(s),  (13)

where {circumflex over (z)} is a unit normal in the positive vertical (+z) direction and

₂ is the magnetic field strength in Region 2 expressed by Equation (1) above. Equation (13) implies that the electric and magnetic fields specified in Equations (1)-(3) may result in a radial surface current density along the boundary interface, where the radial surface current density can be specified by

J _(ρ)(ρ′)=−AH ₁ ⁽²⁾(−jγρ′)  (14)

where A is a constant. Further, it should be noted that close-in to the guided surface waveguide probe 200 (for ρ<<λ), Equation (14) above has the behavior

$\begin{matrix} {{J_{close}\left( \rho^{\prime} \right)} = {\frac{- {A\left( {j\; 2} \right)}}{\pi \left( {{- \; j}\; \gamma \; \rho^{\prime}} \right)} = {{- H_{\varphi}} = {- {\frac{I_{0}}{2{\pi\rho}^{\prime}}.}}}}} & (15) \end{matrix}$

The negative sign means that when source current (I_(o)) flows vertically upward as illustrated in FIG. 3, the “close-in” ground current flows radially inward. By field matching on H_(ϕ)“close-in,” it can be determined that

$\begin{matrix} {A = {{- \; \frac{I_{o}\gamma}{4}} = {- \frac{\omega \; q_{1}\gamma}{4}}}} & (16) \end{matrix}$

where q₁=C₁V₁, in Equations (1)-(6) and (14). Therefore, the radial surface current density of Equation (14) can be restated as

$\begin{matrix} {{J_{\rho}\left( \rho^{\prime} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho^{\prime}} \right)}.}}} & (17) \end{matrix}$

The fields expressed by Equations (1)-(6) and (17) have the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation. See Barlow, H. M. and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankel functions used in Equations (1)-(6) and (17) is provided for these solutions of the wave equation. One might observe that the Hankel functions of the first and second kind and order n are defined as complex combinations of the standard Bessel functions of the first and second kinds

H _(n) ⁽¹⁾(x)=J _(n)(x)+jN _(n)(x), and  (18)

H _(n) ⁽²⁾(x)=J _(n)(x)−jN _(n)(x),  (19)

These functions represent cylindrical waves propagating radially inward (H_(n) ⁽¹⁾) and outward (H_(n) ⁽²⁾), respectively. The definition is analogous to the relationship e^(±jx)=cos x±j sin x. See, for example, Harrington, R. F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That H_(n) ⁽²⁾ (k_(ρ)ρ) is an outgoing wave can be recognized from its large argument asymptotic behavior that is obtained directly from the series definitions of J_(n)(x) and N_(n)(x). Far-out from the guided surface waveguide probe:

$\begin{matrix} {{{{H_{n}^{(2)}(x)}\underset{x\rightarrow\infty}{\rightarrow}{\sqrt{\frac{2\; j}{\pi \; x}}j^{n}e^{- {jx}}}} = {\sqrt{\frac{2}{\pi \; x}}j^{n}e^{- {j{({x - \frac{\pi}{4}})}}}}},} & \left( {20a} \right) \end{matrix}$

which, when multiplied by e^(jωt), is an outward propagating cylindrical wave of the form e^(j(ωt−kρ)) with a 1/√{square root over (ρ)} spatial variation. The first order (n=1) solution can be determined from Equation (20a) to be

$\begin{matrix} {{{H_{1}^{(2)}(x)}\underset{x\rightarrow\infty}{\rightarrow}{j\sqrt{\frac{2\; j}{\pi \; x}}e^{{- j}\; x}}} = {\sqrt{\frac{2}{\pi \; x}}{e^{- {j{({x - \frac{\pi}{2} - \frac{\pi}{4}})}}}.}}} & \left( {20b} \right) \end{matrix}$

Close-in to the guided surface waveguide probe (for ρ<<λ), the Hankel function of first order and the second kind behaves as

$\begin{matrix} {{H_{1}^{(2)}(x)}\underset{x\rightarrow 0}{\rightarrow}{\frac{2\; j}{\pi \; x}.}} & (21) \end{matrix}$

Note that these asymptotic expressions are complex quantities. When x is a real quantity, Equations (20b) and (21) differ in phase by √{square root over (j)}, which corresponds to an extra phase advance or “phase boost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotes of the first order Hankel function of the second kind have a Hankel “crossover” or transition point where they are of equal magnitude at a distance of ρ=R_(x).

Thus, beyond the Hankel crossover point the “far out” representation predominates over the “close-in” representation of the Hankel function. The distance to the Hankel crossover point (or Hankel crossover distance) can be found by equating Equations (20b) and (21) for −jγρ, and solving for R_(x). With x=σ/ωε_(o), it can be seen that the far-out and close-in Hankel function asymptotes are frequency dependent, with the Hankel crossover point moving out as the frequency is lowered. It should also be noted that the Hankel function asymptotes may also vary as the conductivity (a) of the lossy conducting medium changes. For example, the conductivity of the soil can vary with changes in weather conditions.

Referring to FIG. 4, shown is an example of a plot of the magnitudes of the first order Hankel functions of Equations (20b) and (21) for a Region 1 conductivity of σ=0.010 mhos/m and relative permittivity ε_(r)=15, at an operating frequency of 1850 kHz. Curve 115 is the magnitude of the far-out asymptote of Equation (20b) and curve 118 is the magnitude of the close-in asymptote of Equation (21), with the Hankel crossover point 121 occurring at a distance of R_(x)=54 feet. While the magnitudes are equal, a phase offset exists between the two asymptotes at the Hankel crossover point 121. It can also be seen that the Hankel crossover distance is much less than a wavelength of the operation frequency.

Considering the electric field components given by Equations (2) and (3) of the Zenneck closed-form solution in Region 2, it can be seen that the ratio of ε_(z) and ε_(ρ) asymptotically passes to

$\begin{matrix} {{\frac{E_{z}}{E_{\rho}} = {{{\left( \frac{{- j}\; \gamma}{u_{2}} \right)\frac{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}\underset{\rho\rightarrow\infty}{\rightarrow}\sqrt{ɛ_{r} - {j\; \frac{\sigma}{\omega \; ɛ_{0}}}}} = {n = {\tan \; \theta_{i}}}}},} & (22) \end{matrix}$

where n is the complex index of refraction of Equation (10) and θ_(i) is the angle of incidence of the electric field. In addition, the vertical component of the mode-matched electric field of Equation (3) asymptotically passes to

$\begin{matrix} {{E_{2\; z}\underset{\rho\rightarrow\infty}{\rightarrow}{\left( \frac{q_{free}}{ɛ_{0}} \right)\sqrt{\frac{\gamma^{3}}{8\; \pi}}e^{{- u_{2}}z}\frac{e^{- {j{({{\gamma \; \rho} - {\pi/4}})}}}}{\sqrt{\rho}}}},} & (23) \end{matrix}$

which is linearly proportional to free charge on the isolated component of the elevated charge terminal's capacitance at the terminal voltage, q_(free)=C_(free)×V_(T).

For example, the height H₁ of the elevated charge terminal T₁ in FIG. 3 affects the amount of free charge on the charge terminal T₁. When the charge terminal T₁ is near the ground plane of Region 1, most of the charge Q₁ on the terminal is “bound.” As the charge terminal T₁ is elevated, the bound charge is lessened until the charge terminal T₁ reaches a height at which substantially all of the isolated charge is free.

The advantage of an increased capacitive elevation for the charge terminal T₁ is that the charge on the elevated charge terminal T₁ is further removed from the ground plane, resulting in an increased amount of free charge q_(free) to couple energy into the guided surface waveguide mode. As the charge terminal T₁ is moved away from the ground plane, the charge distribution becomes more uniformly distributed about the surface of the terminal. The amount of free charge is related to the self-capacitance of the charge terminal T₁.

For example, the capacitance of a spherical terminal can be expressed as a function of physical height above the ground plane. The capacitance of a sphere at a physical height of h above a perfect ground is given by

C _(elevated sphere)=4πε_(o) a(1+M+M ² +M ³+2M ⁴+3M ⁵+ . . . ),  (24)

where the diameter of the sphere is 2a, and where M=a/2h with h being the height of the spherical terminal. As can be seen, an increase in the terminal height h reduces the capacitance C of the charge terminal. It can be shown that for elevations of the charge terminal T₁ that are at a height of about four times the diameter (4D=8a) or greater, the charge distribution is approximately uniform about the spherical terminal, which can improve the coupling into the guided surface waveguide mode.

In the case of a sufficiently isolated terminal, the self-capacitance of a conductive sphere can be approximated by C=4πε_(o)a, where a is the radius of the sphere in meters, and the self-capacitance of a disk can be approximated by C=8ε_(o)a, where a is the radius of the disk in meters. The charge terminal T₁ can include any shape such as a sphere, a disk, a cylinder, a cone, a torus, a hood, one or more rings, or any other randomized shape or combination of shapes. An equivalent spherical diameter can be determined and used for positioning of the charge terminal

This may be further understood with reference to the example of FIG. 3, where the charge terminal T₁ is elevated at a physical height of h_(p)=H₁ above the lossy conducting medium 203. To reduce the effects of the “bound” charge, the charge terminal T₁ can be positioned at a physical height that is at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal T₁ to reduce the bounded charge effects.

Referring next to FIG. 5A, shown is a ray optics interpretation of the electric field produced by the elevated charge Q₁ on charge terminal T₁ of FIG. 3. As in optics, minimizing the reflection of the incident electric field can improve and/or maximize the energy coupled into the guided surface waveguide mode of the lossy conducting medium 203. For an electric field (E_(∥)) that is polarized parallel to the plane of incidence (not the boundary interface), the amount of reflection of the incident electric field may be determined using the Fresnel reflection coefficient, which can be expressed as

$\begin{matrix} {{{\Gamma_{\parallel}\left( \theta_{i} \right)} = {\frac{E_{\parallel {,R}}}{E_{\parallel {,i}}} = \frac{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {j\; x}} \right)\cos \; \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {j\; x}} \right)\cos \; \theta_{i}}}}},} & (25) \end{matrix}$

where θ_(i) is the conventional angle of incidence measured with respect to the surface normal.

In the example of FIG. 5A, the ray optic interpretation shows the incident field polarized parallel to the plane of incidence having an angle of incidence of θ_(i), which is measured with respect to the surface normal ({circumflex over (z)}). There will be no reflection of the incident electric field when Γ_(∥)(θ_(i))=0 and thus the incident electric field will be completely coupled into a guided surface waveguide mode along the surface of the lossy conducting medium 203. It can be seen that the numerator of Equation (25) goes to zero when the angle of incidence is

θ_(i)=arctan(√{square root over (ε_(r) −jx)})=θ_(i,B),  (26)

where x=σ/ωε_(o). This complex angle of incidence (θ_(i,B)) is referred to as the Brewster angle. Referring back to Equation (22), it can be seen that the same complex Brewster angle (θ_(i,B)) relationship is present in both Equations (22) and (26).

As illustrated in FIG. 5A, the electric field vector E can be depicted as an incoming non-uniform plane wave, polarized parallel to the plane of incidence. The electric field vector E can be created from independent horizontal and vertical components as

(θ_(i))=E _(ρ) {circumflex over (ρ)}+E _(z) {circumflex over (z)}.  (27)

Geometrically, the illustration in FIG. 5A suggests that the electric field vector E can be given by

$\begin{matrix} {{{E_{\rho}\left( {\rho,z} \right)} = {{E\left( {\rho,z} \right)}\cos \; \theta_{i}}},{and}} & \left( {28a} \right) \\ {{{E_{z}\left( {\rho,z} \right)} = {{{E\left( {\rho,z} \right)}{\cos \left( {\frac{\pi}{2} - \theta_{i}} \right)}} = {{E\left( {\rho,z} \right)}\sin \; \theta_{i}}}},} & \left( {28b} \right) \end{matrix}$

which means that the field ratio is

$\begin{matrix} {\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \; \theta_{i}} = {\tan \; {\psi_{i}.}}}} & (29) \end{matrix}$

A generalized parameter W, called “wave tilt,” is noted herein as the ratio of the horizontal electric field component to the vertical electric field component given by

$\begin{matrix} {{W = {\frac{E_{\rho}}{E_{z}} = {{W}e^{j\; \Psi}}}},{or}} & \left( {30a} \right) \\ {{\frac{1}{W} = {\frac{E_{z}}{E_{\rho}} = {{\tan \; \theta_{i}} = {\frac{1}{W}e^{{- j}\; \Psi}}}}},} & \left( {30b} \right) \end{matrix}$

which is complex and has both magnitude and phase. For an electromagnetic wave in Region 2, the wave tilt angle (Ψ) is equal to the angle between the normal of the wave-front at the boundary interface with Region 1 and the tangent to the boundary interface. This may be easier to see in FIG. 5B, which illustrates equi-phase surfaces of an electromagnetic wave and their normals for a radial cylindrical guided surface wave. At the boundary interface (z=0) with a perfect conductor, the wave-front normal is parallel to the tangent of the boundary interface, resulting in W=0. However, in the case of a lossy dielectric, a wave tilt W exists because the wave-front normal is not parallel with the tangent of the boundary interface at z=0.

Applying Equation (30b) to a guided surface wave gives

$\begin{matrix} {{\tan \; \theta_{i,B}} = {\frac{E_{z}}{E_{\rho}} = {\frac{u_{2}}{\gamma} = {\sqrt{ɛ_{r} - {j\; x}} = {n = {\frac{1}{W} = {\frac{1}{W}{e^{{- j}\; \Psi}.}}}}}}}} & (31) \end{matrix}$

With the angle of incidence equal to the complex Brewster angle (θ_(i,B)), the Fresnel reflection coefficient of Equation (25) vanishes, as shown by

$\begin{matrix} {{\Gamma_{\parallel}\left( \theta_{i,B} \right)} = {{\frac{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\; \theta_{i}}} - {\left( {ɛ_{r} - {j\; x}} \right)\cos \; \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\; \theta_{i}}} + {\left( {ɛ_{r} - {j\; x}} \right)\cos \; \theta_{i}}}_{\theta_{i} = \theta_{i,B}}} = 0.}} & (32) \end{matrix}$

By adjusting the complex field ratio of Equation (22), an incident field can be synthesized to be incident at a complex angle at which the reflection is reduced or eliminated. Establishing this ratio as n=√{square root over (ε_(r)−jx)} results in the synthesized electric field being incident at the complex Brewster angle, making the reflections vanish.

The concept of an electrical effective height can provide further insight into synthesizing an electric field with a complex angle of incidence with a guided surface waveguide probe 200. The electrical effective height (h_(eff)) has been defined as

$\begin{matrix} {h_{eff} = {\frac{1}{I_{0}}{\int_{0}^{h_{p}}{{I(z)}{dz}}}}} & (33) \end{matrix}$

for a monopole with a physical height (or length) of h_(p). Since the expression depends upon the magnitude and phase of the source distribution along the structure, the effective height (or length) is complex in general. The integration of the distributed current I(z) of the structure is performed over the physical height of the structure (h_(p)), and normalized to the ground current (I₀) flowing upward through the base (or input) of the structure. The distributed current along the structure can be expressed by

I(z)=I _(C) cos(β₀ z),  (34)

where β₀ is the propagation factor for current propagating on the structure. In the example of FIG. 3, I_(C) is the current that is distributed along the vertical structure of the guided surface waveguide probe 200 a.

For example, consider a feed network 209 that includes a low loss coil (e.g., a helical coil) at the bottom of the structure and a vertical feed line conductor connected between the coil and the charge terminal T₁. The phase delay due to the coil (or helical delay line) is θ_(c)=β_(p)l_(C), with a physical length of l_(C) and a propagation factor of

$\begin{matrix} {{\beta_{p} = {\frac{2\; \pi}{\lambda_{p}} = \frac{2\; \pi}{V_{f}\lambda_{0}}}},} & (35) \end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is the wavelength at the supplied frequency, and λ_(p) is the propagation wavelength resulting from the velocity factor V_(f). The phase delay is measured relative to the ground (stake) current I₀.

In addition, the spatial phase delay along the length l_(w) of the vertical feed line conductor can be given by θ_(y)=β_(w)l_(w) where β_(w) is the propagation phase constant for the vertical feed line conductor. In some implementations, the spatial phase delay may be approximated by θ_(y)=β_(w)h_(p) since the difference between the physical height h_(p) of the guided surface waveguide probe 200 a and the vertical feed line conductor length l_(w) is much less than a wavelength at the supplied frequency (λ₀). As a result, the total phase delay through the coil and vertical feed line conductor is Φ=θ_(c)+θ_(y), and the current fed to the top of the coil from the bottom of the physical structure is

I _(C)(θ_(c)+θ_(y))=I ₀ e ^(jΦ),  (36)

with the total phase delay Φ measured relative to the ground (stake) current I₀. Consequently, the electrical effective height of a guided surface waveguide probe 200 can be approximated by

$\begin{matrix} {{h_{eff} = {{\frac{1}{I_{0}}{\int_{0}^{h_{p}}{I_{0}e^{j\; \Phi}{\cos \left( {\beta_{0}z} \right)}{dz}}}} \cong {h_{p}e^{j\; \Phi}}}},} & (37) \end{matrix}$

for the case where the physical height h_(p)<<λ₀. The complex effective height of a monopole, h_(eff)=h_(p) at an angle (or phase shift) of Φ, may be adjusted to cause the source fields to match a guided surface waveguide mode and cause a guided surface wave to be launched on the lossy conducting medium 203.

In the example of FIG. 5A, ray optics are used to illustrate the complex angle trigonometry of the incident electric field (E) having a complex Brewster angle of incidence (θ_(i,B)) at the Hankel crossover distance (R_(x)) 121. Recall from Equation (26) that, for a lossy conducting medium, the Brewster angle is complex and specified by

$\begin{matrix} {{\tan \mspace{11mu} \theta_{i,B}} = {\sqrt{ɛ_{r} - {j\frac{\sigma}{{\omega ɛ}_{o}}}} = {n.}}} & (38) \end{matrix}$

Electrically, the geometric parameters are related by the electrical effective height (h_(eff)) of the charge terminal T₁ by

R _(x) tan ψ_(i,B) =R _(x) ×W=h _(eff) =h _(p) e ^(jΦ),  (39)

where ψ_(i,B)=(π/2)−θ_(i,B) is the Brewster angle measured from the surface of the lossy conducting medium. To couple into the guided surface waveguide mode, the wave tilt of the electric field at the Hankel crossover distance can be expressed as the ratio of the electrical effective height and the Hankel crossover distance

$\begin{matrix} {\frac{h_{eff}}{R_{x}} = {{\tan \mspace{14mu} \psi_{i,B}} = {W_{Rx}.}}} & (40) \end{matrix}$

Since both the physical height (h_(p)) and the Hankel crossover distance (R_(x)) are real quantities, the angle (Ψ) of the desired guided surface wave tilt at the Hankel crossover distance (R_(x)) is equal to the phase (Φ) of the complex effective height (h_(eff)). This implies that by varying the phase at the supply point of the coil, and thus the phase shift in Equation (37), the phase, Φ, of the complex effective height can be manipulated to match the angle of the wave tilt, Ψ, of the guided surface waveguide mode at the Hankel crossover point 121: Φ=Ψ.

In FIG. 5A, a right triangle is depicted having an adjacent side of length R_(x) along the lossy conducting medium surface and a complex Brewster angle ψ_(i,B) measured between a ray 124 extending between the Hankel crossover point 121 at R_(x) and the center of the charge terminal T₁, and the lossy conducting medium surface 127 between the Hankel crossover point 121 and the charge terminal T₁. With the charge terminal T₁ positioned at physical height h_(p) and excited with a charge having the appropriate phase delay Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R_(x), and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection.

If the physical height of the charge terminal T₁ is decreased without changing the phase shift Φ of the effective height (h_(eff)), the resulting electric field intersects the lossy conducting medium 203 at the Brewster angle at a reduced distance from the guided surface waveguide probe 200. FIG. 6 graphically illustrates the effect of decreasing the physical height of the charge terminal T₁ on the distance where the electric field is incident at the Brewster angle. As the height is decreased from h₃ through h₂ to h₁, the point where the electric field intersects with the lossy conducting medium (e.g., the Earth) at the Brewster angle moves closer to the charge terminal position. However, as Equation (39) indicates, the height H₁ (FIG. 3) of the charge terminal T₁ should be at or higher than the physical height (h_(p)) in order to excite the far-out component of the Hankel function. With the charge terminal T₁ positioned at or above the effective height (h_(eff)), the lossy conducting medium 203 can be illuminated at the Brewster angle of incidence (ψ_(i,B)=(π/2)−θ_(i,B)) at or beyond the Hankel crossover distance (R_(x)) 121 as illustrated in FIG. 5A. To reduce or minimize the bound charge on the charge terminal T₁, the height should be at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal T₁ as mentioned above.

A guided surface waveguide probe 200 can be configured to establish an electric field having a wave tilt that corresponds to a wave illuminating the surface of the lossy conducting medium 203 at a complex Brewster angle, thereby exciting radial surface currents by substantially mode-matching to a guided surface wave mode at (or beyond) the Hankel crossover point 121 at R_(x).

Referring to FIG. 7, shown is a graphical representation of an example of a guided surface waveguide probe 200 b that includes a charge terminal T₁. An AC source 212 acts as the excitation source for the charge terminal T₁, which is coupled to the guided surface waveguide probe 200 b through a feed network 209 (FIG. 3) comprising a coil 215 such as, e.g., a helical coil. In other implementations, the AC source 212 can be inductively coupled to the coil 215 through a primary coil. In some embodiments, an impedance matching network may be included to improve and/or maximize coupling of the AC source 212 to the coil 215.

As shown in FIG. 7, the guided surface waveguide probe 200 b can include the upper charge terminal T₁ (e.g., a sphere at height h_(p)) that is positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. A second medium 206 is located above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C_(T). During operation, charge Q₁ is imposed on the terminal T₁ depending on the voltage applied to the terminal T₁ at any given instant.

In the example of FIG. 7, the coil 215 is coupled to a ground stake 218 at a first end and to the charge terminal T₁ via a vertical feed line conductor 221. In some implementations, the coil connection to the charge terminal T₁ can be adjusted using a tap 224 of the coil 215 as shown in FIG. 7. The coil 215 can be energized at an operating frequency by the AC source 212 through a tap 227 at a lower portion of the coil 215. In other implementations, the AC source 212 can be inductively coupled to the coil 215 through a primary coil.

The construction and adjustment of the guided surface waveguide probe 200 is based upon various operating conditions, such as the transmission frequency, conditions of the lossy conducting medium (e.g., soil conductivity a and relative permittivity ε_(r)), and size of the charge terminal T₁. The index of refraction can be calculated from Equations (10) and (11) as

n=√{square root over (ε_(r) −jx)},  (41)

where x=σ/ωε_(o) with ω=2πf. The conductivity a and relative permittivity ε_(r) can be determined through test measurements of the lossy conducting medium 203. The complex Brewster angle (θ_(i,B)) measured from the surface normal can also be determined from Equation (26) as

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)}),  (42)

or measured from the surface as shown in FIG. 5A as

$\begin{matrix} {\psi_{i,B} = {\frac{\pi}{2} - {\theta_{i,B}.}}} & (43) \end{matrix}$

The wave tilt at the Hankel crossover distance (W_(Rx)) can also be found using Equation (40).

The Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for −jγρ, and solving for R_(x) as illustrated by FIG. 4. The electrical effective height can then be determined from Equation (39) using the Hankel crossover distance and the complex Brewster angle as

h _(eff) =h _(p) e ^(jΦ) =R _(x) tan ψ_(i,B).  (44)

As can be seen from Equation (44), the complex effective height (h_(eff)) includes a magnitude that is associated with the physical height (h_(p)) of the charge terminal T₁ and a phase delay (Φ) that is to be associated with the angle (Ψ) of the wave tilt at the Hankel crossover distance (R_(x)). With these variables and the selected charge terminal T₁ configuration, it is possible to determine the configuration of a guided surface waveguide probe 200.

With the charge terminal T₁ positioned at or above the physical height (h_(p)), the feed network 209 (FIG. 3) and/or the vertical feed line connecting the feed network to the charge terminal T₁ can be adjusted to match the phase (Φ) of the charge Q₁ on the charge terminal T₁ to the angle (Ψ) of the wave tilt (W). The size of the charge terminal T₁ can be chosen to provide a sufficiently large surface for the charge Q₁ imposed on the terminals. In general, it is desirable to make the charge terminal T₁ as large as practical. The size of the charge terminal T₁ should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal.

The phase delay θ_(c) of a helically-wound coil can be determined from Maxwell's equations as has been discussed by Corum, K. L. and J. F. Corum, “RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes,” Microwave Review, Vol. 7, No. 2, September 2001, pp. 36-45., which is incorporated herein by reference in its entirety. For a helical coil with H/D>1, the ratio of the velocity of propagation (v) of a wave along the coil's longitudinal axis to the speed of light (c), or the “velocity factor,” is given by

$\begin{matrix} {{V_{f} = {\frac{\upsilon}{c} = \frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}}}},} & (45) \end{matrix}$

where H is the axial length of the solenoidal helix, D is the coil diameter, N is the number of turns of the coil, s=H/N is the turn-to-turn spacing (or helix pitch) of the coil, and λ_(o) is the free-space wavelength. Based upon this relationship, the electrical length, or phase delay, of the helical coil is given by

$\begin{matrix} {\theta_{c} = {{\beta_{p}H} = {{\frac{2\pi}{\lambda}H} = {\frac{2\pi}{V_{f}\lambda_{o}}{H.}}}}} & (46) \end{matrix}$

The principle is the same if the helix is wound spirally or is short and fat, but V_(f) and θ_(c) are easier to obtain by experimental measurement. The expression for the characteristic (wave) impedance of a helical transmission line has also been derived as

$\begin{matrix} {Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\; {n\left( \frac{V_{f}\lambda_{o}}{D} \right)}} - 1.027} \right\rbrack}.}} & (47) \end{matrix}$

The spatial phase delay θ_(y) of the structure can be determined using the traveling wave phase delay of the vertical feed line conductor 221 (FIG. 7). The capacitance of a cylindrical vertical conductor above a prefect ground plane can be expressed as

$\begin{matrix} {{C_{A} = {\frac{2{\pi ɛ}_{o}h_{w}}{{\; {n\left( \frac{h}{a} \right)}} - 1}{Farads}}},} & (48) \end{matrix}$

where h_(w) is the vertical length (or height) of the conductor and a is the radius (in mks units). As with the helical coil, the traveling wave phase delay of the vertical feed line conductor can be given by

$\begin{matrix} {{\theta_{y} = {{\beta_{w}h_{w}} = {{\frac{2\pi}{\lambda_{w}}h_{w}} = {\frac{2\pi}{V_{w}\lambda_{o}}h_{w}}}}},} & (49) \end{matrix}$

where β_(w) is the propagation phase constant for the vertical feed line conductor, h_(w) is the vertical length (or height) of the vertical feed line conductor, V_(w) is the velocity factor on the wire, λ₀ is the wavelength at the supplied frequency, and λ_(w) is the propagation wavelength resulting from the velocity factor V_(w). For a uniform cylindrical conductor, the velocity factor is a constant with V_(w)≈0.94, or in a range from about 0.93 to about 0.98. If the mast is considered to be a uniform transmission line, its average characteristic impedance can be approximated by

$\begin{matrix} {{Z_{w} = {\frac{60}{V_{w}}\left\lbrack {{\; {n\left( \frac{h_{w}}{a} \right)}} - 1} \right\rbrack}},} & (50) \end{matrix}$

where V_(w)≈0.94 for a uniform cylindrical conductor and a is the radius of the conductor. An alternative expression that has been employed in amateur radio literature for the characteristic impedance of a single-wire feed line can be given by

$\begin{matrix} {Z_{w} = {138\mspace{14mu} {{\log \left( \frac{1.123\mspace{14mu} V_{w}{\lambda 0}}{2\pi \; a} \right)}.}}} & (51) \end{matrix}$

Equation (51) implies that Z_(w) for a single-wire feeder varies with frequency. The phase delay can be determined based upon the capacitance and characteristic impedance.

With a charge terminal T₁ positioned over the lossy conducting medium 203 as shown in FIG. 3, the feed network 209 can be adjusted to excite the charge terminal T₁ with the phase shift (Φ) of the complex effective height (h_(eff)) equal to the angle (Ψ) of the wave tilt at the Hankel crossover distance, or Φ=Ψ. When this condition is met, the electric field produced by the charge oscillating Q₁ on the charge terminal T₁ is coupled into a guided surface waveguide mode traveling along the surface of a lossy conducting medium 203. For example, if the Brewster angle (θ_(i,B)), the phase delay (θ_(y)) associated with the vertical feed line conductor 221 (FIG. 7), and the configuration of the coil 215 (FIG. 7) are known, then the position of the tap 224 (FIG. 7) can be determined and adjusted to impose an oscillating charge Q₁ on the charge terminal T₁ with phase Φ=Ψ. The position of the tap 224 may be adjusted to maximize coupling the traveling surface waves into the guided surface waveguide mode. Excess coil length beyond the position of the tap 224 can be removed to reduce the capacitive effects. The vertical wire height and/or the geometrical parameters of the helical coil may also be varied.

The coupling to the guided surface waveguide mode on the surface of the lossy conducting medium 203 can be improved and/or optimized by tuning the guided surface waveguide probe 200 for standing wave resonance with respect to a complex image plane associated with the charge Q₁ on the charge terminal T₁. By doing this, the performance of the guided surface waveguide probe 200 can be adjusted for increased and/or maximum voltage (and thus charge Q₁) on the charge terminal T₁. Referring back to FIG. 3, the effect of the lossy conducting medium 203 in Region 1 can be examined using image theory analysis.

Physically, an elevated charge Q₁ placed over a perfectly conducting plane attracts the free charge on the perfectly conducting plane, which then “piles up” in the region under the elevated charge Q₁. The resulting distribution of “bound” electricity on the perfectly conducting plane is similar to a bell-shaped curve. The superposition of the potential of the elevated charge Q₁, plus the potential of the induced “piled up” charge beneath it, forces a zero equipotential surface for the perfectly conducting plane. The boundary value problem solution that describes the fields in the region above the perfectly conducting plane may be obtained using the classical notion of image charges, where the field from the elevated charge is superimposed with the field from a corresponding “image” charge below the perfectly conducting plane.

This analysis may also be used with respect to a lossy conducting medium 203 by assuming the presence of an effective image charge Q₁′ beneath the guided surface waveguide probe 200. The effective image charge Q₁′ coincides with the charge Q₁ on the charge terminal T₁ about a conducting image ground plane 130, as illustrated in FIG. 3. However, the image charge Q₁′ is not merely located at some real depth and 180° out of phase with the primary source charge Q₁ on the charge terminal T₁, as they would be in the case of a perfect conductor. Rather, the lossy conducting medium 203 (e.g., a terrestrial medium) presents a phase shifted image. That is to say, the image charge Q₁′ is at a complex depth below the surface (or physical boundary) of the lossy conducting medium 203. For a discussion of complex image depth, reference is made to Wait, J. R., “Complex Image Theory—Revisited,” IEEE Antennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp. 27-29, which is incorporated herein by reference in its entirety.

Instead of the image charge Q₁′ being at a depth that is equal to the physical height (H₁) of the charge Q₁, the conducting image ground plane 130 (representing a perfect conductor) is located at a complex depth of z=−d/2 and the image charge Q₁′ appears at a complex depth (i.e., the “depth” has both magnitude and phase), given by −D₁=−(d/2+d/2+H₁)≠H₁. For vertically polarized sources over the Earth,

$\begin{matrix} {{d = {{\frac{2\sqrt{\gamma_{e}^{2} + k_{0}^{2}}}{\gamma_{e}^{2}} \approx \frac{2}{\gamma_{e}}} = {{d_{r} + {jd}_{i}} = {{d}{\angle\zeta}}}}},} & (52) \\ {{\gamma_{e}^{2} = {{j\; {\omega\mu}_{1}\sigma_{1}} - {\omega^{2}\mu_{1}ɛ_{1}}}},{and}} & (53) \\ {{k_{o} = {\omega \sqrt{\mu_{o}ɛ_{o}}}},} & (54) \end{matrix}$

as indicated in Equation (12). The complex spacing of the image charge, in turn, implies that the external field will experience extra phase shifts not encountered when the interface is either a dielectric or a perfect conductor. In the lossy conducting medium, the wave front normal is parallel to the tangent of the conducting image ground plane 130 at z=−d/2, and not at the boundary interface between Regions 1 and 2.

Consider the case illustrated in FIG. 8A where the lossy conducting medium 203 is a finitely conducting Earth 133 with a physical boundary 136. The finitely conducting Earth 133 may be replaced by a perfectly conducting image ground plane 139 as shown in FIG. 8B, which is located at a complex depth z₁ below the physical boundary 136. This equivalent representation exhibits the same impedance when looking down into the interface at the physical boundary 136. The equivalent representation of FIG. 8B can be modeled as an equivalent transmission line, as shown in FIG. 8C. The cross-section of the equivalent structure is represented as a (z-directed) end-loaded transmission line, with the impedance of the perfectly conducting image plane being a short circuit (z_(s)=0). The depth z₁ can be determined by equating the TEM wave impedance looking down at the Earth to an image ground plane impedance z_(in) seen looking into the transmission line of FIG. 8C.

In the case of FIG. 8A, the propagation constant and wave intrinsic impedance in the upper region (air) 142 are

$\begin{matrix} {{\gamma_{o} = {{j\; \omega \sqrt{\mu_{o}ɛ_{o}}} = {0 + {j\; \beta_{o}}}}},{and}} & (55) \\ {z_{o} = {\frac{j\; {\omega\mu}_{o}}{\gamma_{o}} = {\sqrt{\frac{\mu_{o}}{ɛ_{o}}}.}}} & (56) \end{matrix}$

In the lossy Earth 133, the propagation constant and wave intrinsic impedance are

$\begin{matrix} {{\gamma_{e} = \sqrt{j\; {{\omega\mu}_{1}\left( {\sigma_{1} + {j\; {\omega ɛ}_{1}}} \right)}}},{and}} & (57) \\ {Z_{e} = {\frac{j\; {\omega\mu}_{1}}{\gamma_{e}}.}} & (58) \end{matrix}$

For normal incidence, the equivalent representation of FIG. 8B is equivalent to a TEM transmission line whose characteristic impedance is that of air (z_(o)), with propagation constant of γ_(o), and whose length is z₁. As such, the image ground plane impedance Z_(in) seen at the interface for the shorted transmission line of FIG. 8C is given by

Z _(in) =Z _(o) tan h(γ_(o) z ₁).  (59)

Equating the image ground plane impedance Z_(in) associated with the equivalent model of FIG. 8C to the normal incidence wave impedance of FIG. 8A and solving for z₁ gives the distance to a short circuit (the perfectly conducting image ground plane 139) as

$\begin{matrix} {{z_{1} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{Z_{e}}{Z_{o}} \right)}} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{\gamma_{o}}{\gamma_{e}} \right)}} \approx \frac{1}{\gamma_{e}}}}},} & (60) \end{matrix}$

where only the first term of the series expansion for the inverse hyperbolic tangent is considered for this approximation. Note that in the air region 142, the propagation constant is γ_(o)=jβ_(o), so Z_(in)=jZ_(o) tan β_(o)z₁ (which is a purely imaginary quantity for a real z₁), but z_(e) is a complex value if σ≠0. Therefore, Z_(in)=Z_(e) only when z₁ is a complex distance.

Since the equivalent representation of FIG. 8B includes a perfectly conducting image ground plane 139, the image depth for a charge or current lying at the surface of the Earth (physical boundary 136) is equal to distance z₁ on the other side of the image ground plane 139, or d=2× z₁ beneath the Earth's surface (which is located at z=0). Thus, the distance to the perfectly conducting image ground plane 139 can be approximated by

$\begin{matrix} {d = {{2\; z_{1}} \approx {\frac{2}{\gamma_{e}}.}}} & (61) \end{matrix}$

Additionally, the “image charge” will be “equal and opposite” to the real charge, so the potential of the perfectly conducting image ground plane 139 at depth z₁=−d/2 will be zero.

If a charge Q₁ is elevated a distance H₁ above the surface of the Earth as illustrated in FIG. 3, then the image charge Q₁′ resides at a complex distance of D₁=d+H₁ below the surface, or a complex distance of d/2+H₁ below the image ground plane 130. The guided surface waveguide probe 200 b of FIG. 7 can be modeled as an equivalent single-wire transmission line image plane model that can be based upon the perfectly conducting image ground plane 139 of FIG. 8B. FIG. 9A shows an example of the equivalent single-wire transmission line image plane model, and FIG. 9B illustrates an example of the equivalent classic transmission line model, including the shorted transmission line of FIG. 8C.

In the equivalent image plane models of FIGS. 9A and 9B, Φ=θ_(y) θ_(c) is the traveling wave phase delay of the guided surface waveguide probe 200 referenced to Earth 133 (or the lossy conducting medium 203), θ_(c)=β_(p)H is the electrical length of the coil 215 (FIG. 7), of physical length H, expressed in degrees, θ_(y)=β_(w)h_(w) is the electrical length of the vertical feed line conductor 221 (FIG. 7), of physical length h_(w), expressed in degrees, and θ_(d)=β_(o) d/2 is the phase shift between the image ground plane 139 and the physical boundary 136 of the Earth 133 (or lossy conducting medium 203). In the example of FIGS. 9A and 9B, Z_(w) is the characteristic impedance of the elevated vertical feed line conductor 221 in ohms, Z_(c) is the characteristic impedance of the coil 215 in ohms, and Z_(O) is the characteristic impedance of free space.

At the base of the guided surface waveguide probe 200, the impedance seen “looking up” into the structure is Z_(↑)=Z_(base). With a load impedance of:

$\begin{matrix} {{Z_{L} = \frac{1}{j\; {\omega C}_{T}}},} & (62) \end{matrix}$

where C_(T) is the self-capacitance of the charge terminal T₁, the impedance seen “looking up” into the vertical feed line conductor 221 (FIG. 7) is given by:

$\begin{matrix} {{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{w} + {Z_{L}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{L}{\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (63) \end{matrix}$

and the impedance seen “looking up” into the coil 215 (FIG. 7) is given by:

$\begin{matrix} {Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}.}}}} & (64) \end{matrix}$

At the base of the guided surface waveguide probe 200, the impedance seen “looking down” into the lossy conducting medium 203 is Z_(↓)=Z_(in), which is given by:

$\begin{matrix} {{Z_{in} = {{Z_{o}\frac{Z_{s} + {Z_{o}{\tanh \left\lbrack {j\; {\beta_{o}\left( {d/2} \right)}} \right\rbrack}}}{Z_{o} + {Z_{s}{\tanh \left\lbrack {j\; {\beta_{o}\left( {d/2} \right)}} \right\rbrack}}}} = {Z_{o}{\tanh \left( {j\; \theta_{d}} \right)}}}},} & (65) \end{matrix}$

where Z_(s)=0.

Neglecting losses, the equivalent image plane model can be tuned to resonance when Z_(↓)+Z_(↑)=0 at the physical boundary 136. Or, in the low loss case, X_(↓)+X_(↑)=0 at the physical boundary 136, where X is the corresponding reactive component. Thus, the impedance at the physical boundary 136 “looking up” into the guided surface waveguide probe 200 is the conjugate of the impedance at the physical boundary 136 “looking down” into the lossy conducting medium 203. By adjusting the load impedance Z_(L) of the charge terminal T₁ while maintaining the traveling wave phase delay Φ equal to the angle of the media's wave tilt Ψ, so that Φ=Ψ, which improves and/or maximizes coupling of the probe's electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth), the equivalent image plane models of FIGS. 9A and 9B can be tuned to resonance with respect to the image ground plane 139. In this way, the impedance of the equivalent complex image plane model is purely resistive, which maintains a superposed standing wave on the probe structure that maximizes the voltage and elevated charge on terminal T₁, and by equations (1)-(3) and (16) maximizes the propagating surface wave.

It follows from the Hankel solutions, that the guided surface wave excited by the guided surface waveguide probe 200 is an outward propagating traveling wave. The source distribution along the feed network 209 between the charge terminal T₁ and the ground stake 218 of the guided surface waveguide probe 200 (FIGS. 3 and 7) is actually composed of a superposition of a traveling wave plus a standing wave on the structure. With the charge terminal T₁ positioned at or above the physical height h_(p), the phase delay of the traveling wave moving through the feed network 209 is matched to the angle of the wave tilt associated with the lossy conducting medium 203. This mode-matching allows the traveling wave to be launched along the lossy conducting medium 203. Once the phase delay has been established for the traveling wave, the load impedance Z_(L) of the charge terminal T₁ is adjusted to bring the probe structure into standing wave resonance with respect to the image ground plane (130 of FIG. 3 or 139 of FIG. 8), which is at a complex depth of −d/2. In that case, the impedance seen from the image ground plane has zero reactance and the charge on the charge terminal T₁ is maximized.

The distinction between the traveling wave phenomenon and standing wave phenomena is that (1) the phase delay of traveling waves (θ=βd) on a section of transmission line of length d (sometimes called a “delay line”) is due to propagation time delays; whereas (2) the position-dependent phase of standing waves (which are composed of forward and backward propagating waves) depends on both the line length propagation time delay and impedance transitions at interfaces between line sections of different characteristic impedances. In addition to the phase delay that arises due to the physical length of a section of transmission line operating in sinusoidal steady-state, there is an extra reflection coefficient phase at impedance discontinuities that is due to the ratio of Z_(oa)/Z_(ob), where Z_(oa) and Z_(ob) are the characteristic impedances of two sections of a transmission line such as, e.g., a helical coil section of characteristic impedance Z_(oa)=Z_(c) (FIG. 9B) and a straight section of vertical feed line conductor of characteristic impedance Z_(ob)=Z_(w) (FIG. 9B).

As a result of this phenomenon, two relatively short transmission line sections of widely differing characteristic impedance may be used to provide a very large phase shift. For example, a probe structure composed of two sections of transmission line, one of low impedance and one of high impedance, together totaling a physical length of, say, 0.05λ, may be fabricated to provide a phase shift of 90° which is equivalent to a 0.25λ resonance. This is due to the large jump in characteristic impedances. In this way, a physically short probe structure can be electrically longer than the two physical lengths combined. This is illustrated in FIGS. 9A and 9B, where the discontinuities in the impedance ratios provide large jumps in phase. The impedance discontinuity provides a substantial phase shift where the sections are joined together.

Referring to FIG. 10, shown is a flow chart 150 illustrating an example of adjusting a guided surface waveguide probe 200 (FIGS. 3 and 7) to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium, which launches a guided surface traveling wave along the surface of a lossy conducting medium 203 (FIG. 3). Beginning with 153, the charge terminal T₁ of the guided surface waveguide probe 200 is positioned at a defined height above a lossy conducting medium 203. Utilizing the characteristics of the lossy conducting medium 203 and the operating frequency of the guided surface waveguide probe 200, the Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for −jγρ, and solving for R_(x) as illustrated by FIG. 4. The complex index of refraction (n) can be determined using Equation (41), and the complex Brewster angle (θ_(i,B)) can then be determined from Equation (42). The physical height (h_(p)) of the charge terminal T₁ can then be determined from Equation (44). The charge terminal T₁ should be at or higher than the physical height (h_(p)) in order to excite the far-out component of the Hankel function. This height relationship is initially considered when launching surface waves. To reduce or minimize the bound charge on the charge terminal T₁, the height should be at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal T₁.

At 156, the electrical phase delay Φ of the elevated charge Q₁ on the charge terminal T₁ is matched to the complex wave tilt angle T. The phase delay (θ_(c)) of the helical coil and/or the phase delay (θ_(y)) of the vertical feed line conductor can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). Based on Equation (31), the angle (Ψ) of the wave tilt can be determined from:

$\begin{matrix} {W = {\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \; \theta_{i,B}} = {\frac{1}{n} = {{W}{e^{j\; \Psi}.}}}}}} & (66) \end{matrix}$

The electrical phase Φ can then be matched to the angle of the wave tilt. This angular (or phase) relationship is next considered when launching surface waves. For example, the electrical phase delay Φ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters of the coil 215 (FIG. 7) and/or the length (or height) of the vertical feed line conductor 221 (FIG. 7). By matching Φ=Ψ, an electric field can be established at or beyond the Hankel crossover distance (R_(x)) with a complex Brewster angle at the boundary interface to excite the surface waveguide mode and launch a traveling wave along the lossy conducting medium 203.

Next at 159, the load impedance of the charge terminal T₁ is tuned to resonate the equivalent image plane model of the guided surface waveguide probe 200. The depth (d/2) of the conducting image ground plane 139 of FIG. 9A and 9B (or 130 of FIG. 3) can be determined using Equations (52), (53) and (54) and the values of the lossy conducting medium 203 (e.g., the Earth), which can be measured. Using that depth, the phase shift (θ_(d)) between the image ground plane 139 and the physical boundary 136 of the lossy conducting medium 203 can be determined using θ_(d)=β_(o) d/2. The impedance (Z_(in)) as seen “looking down” into the lossy conducting medium 203 can then be determined using Equation (65). This resonance relationship can be considered to maximize the launched surface waves.

Based upon the adjusted parameters of the coil 215 and the length of the vertical feed line conductor 221, the velocity factor, phase delay, and impedance of the coil 215 and vertical feed line conductor 221 can be determined using Equations (45) through (51). In addition, the self-capacitance (C_(T)) of the charge terminal T₁ can be determined using, e.g., Equation (24). The propagation factor (β_(p)) of the coil 215 can be determined using Equation (35) and the propagation phase constant (β_(w)) for the vertical feed line conductor 221 can be determined using Equation (49). Using the self-capacitance and the determined values of the coil 215 and vertical feed line conductor 221, the impedance (Z_(base)) of the guided surface waveguide probe 200 as seen “looking up” into the coil 215 can be determined using Equations (62), (63) and (64).

The equivalent image plane model of the guided surface waveguide probe 200 can be tuned to resonance by adjusting the load impedance Z_(L) such that the reactance component X_(base) of Z_(base) cancels out the reactance component X_(in) of Z_(in), or X_(base) X_(in)=0. Thus, the impedance at the physical boundary 136 “looking up” into the guided surface waveguide probe 200 is the conjugate of the impedance at the physical boundary 136 “looking down” into the lossy conducting medium 203. The load impedance Z_(L) can be adjusted by varying the capacitance (C_(T)) of the charge terminal T₁ without changing the electrical phase delay Φ=θ_(c)+θ_(y) of the charge terminal T₁. An iterative approach may be taken to tune the load impedance Z_(L) for resonance of the equivalent image plane model with respect to the conducting image ground plane 139 (or 130). In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth) can be improved and/or maximized.

This may be better understood by illustrating the situation with a numerical example. Consider a guided surface waveguide probe 200 comprising a top-loaded vertical stub of physical height h_(p) with a charge terminal T₁ at the top, where the charge terminal T₁ is excited through a helical coil and vertical feed line conductor at an operational frequency (f_(o)) of 1.85 MHz. With a height (H₁) of 16 feet and the lossy conducting medium 203 (e.g., Earth) having a relative permittivity of ε_(r)=15 and a conductivity of σ₁=0.010 mhos/m, several surface wave propagation parameters can be calculated for f_(o)=1.850 MHz. Under these conditions, the Hankel crossover distance can be found to be R_(x)=54.5 feet with a physical height of h_(p)=5.5 feet, which is well below the actual height of the charge terminal T₁. While a charge terminal height of H₁=5.5 feet could have been used, the taller probe structure reduced the bound capacitance, permitting a greater percentage of free charge on the charge terminal T₁ providing greater field strength and excitation of the traveling wave.

The wave length can be determined as:

$\begin{matrix} {{\lambda_{o} = {\frac{c}{f_{o}} = {162.162\mspace{14mu} {meters}}}},} & (67) \end{matrix}$

where c is the speed of light. The complex index of refraction is:

n=√{square root over (ε_(r) −jx)}=7.529−j6.546,  (68)

from Equation (41), where x=σ₁/ωε_(o) with ω=2πf_(o), and the complex Brewster angle is:

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)})=85.6−j3.744°.  (69)

from Equation (42). Using Equation (66), the wave tilt values can be determined to be:

$\begin{matrix} {W = {\frac{1}{\tan \; \theta_{i,B}} = {\frac{1}{n} = {{{W}e^{j\; \Psi}} = {0.101{e^{j\; 40.614{^\circ}}.}}}}}} & (70) \end{matrix}$

Thus, the helical coil can be adjusted to match Φ=Ψ=40.614°

The velocity factor of the vertical feed line conductor (approximated as a uniform cylindrical conductor with a diameter of 0.27 inches) can be given as V_(w)≈0.93. Since h_(p)<<λ_(o), the propagation phase constant for the vertical feed line conductor can be approximated as:

$\begin{matrix} {\beta_{w} = {\frac{2\; \pi}{\lambda_{w}} = {\frac{2\; \pi}{V_{w}\lambda_{0}} = {0.042\mspace{14mu} {m^{- 1}.}}}}} & (71) \end{matrix}$

From Equation (49) the phase delay of the vertical feed line conductor is:

θ_(y)=ρ_(w) h _(w)≈β_(w) h _(p)=11.640°.  (72)

By adjusting the phase delay of the helical coil so that θ_(c)=28.974°=40.614°−11.640°, Φ will equal Ψ to match the guided surface waveguide mode. To illustrate the relationship between Φ and Ψ, FIG. 11 shows a plot of both over a range of frequencies. As both Φ and Ψ are frequency dependent, it can be seen that their respective curves cross over each other at approximately 1.85 MHz.

For a helical coil having a conductor diameter of 0.0881 inches, a coil diameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches, the velocity factor for the coil can be determined using Equation (45) as:

$\begin{matrix} {{V_{f} = {\frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}} = 0.069}},} & (73) \end{matrix}$

and the propagation factor from Equation (35) is:

$\begin{matrix} {\beta_{p} = {\frac{2\; \pi}{V_{f}\lambda_{0}} = {0.564\mspace{14mu} {m^{- 1}.}}}} & (74) \end{matrix}$

With θ_(c)=28.974°, the axial length of the solenoidal helix (H) can be determined using Equation (46) such that:

$\begin{matrix} {H = {\frac{\theta_{c}}{\beta_{p}} = {35.2732\mspace{14mu} {{inches}.}}}} & (75) \end{matrix}$

This height determines the location on the helical coil where the vertical feed line conductor is connected, resulting in a coil with 8.818 turns (N=H/s).

With the traveling wave phase delay of the coil and vertical feed line conductor adjusted to match the wave tilt angle (Φ=θ_(c)+θ_(y)=Ψ), the load impedance (Z_(L)) of the charge terminal T₁ can be adjusted for standing wave resonance of the equivalent image plane model of the guided surface wave probe 200. From the measured permittivity, conductivity and permeability of the Earth, the radial propagation constant can be determined using Equation (57)

γe=√{square root over (jωu ₁(σ₁ +jωε ₁))}=0.25+j0.292 m⁻¹,  (76)

And the complex depth of the conducting image ground plane can be approximated from Equation (52) as:

$\begin{matrix} {{{d \approx \frac{2}{\gamma_{e}\;}} = {3.364 + {j\mspace{14mu} 3.963\mspace{14mu} {meters}}}},} & (77) \end{matrix}$

with a corresponding phase shift between the conducting image ground plane and the physical boundary of the Earth given by:

θ_(d)=β_(o)(d/2)=4.015−j4.73°.  (78)

Using Equation (65), the impedance seen “looking down” into the lossy conducting medium 203 (i.e., Earth) can be determined as:

Z _(in) =Z _(o) tan h(jθ _(d))=R _(in) +jX _(in)=31.191+j26.27 ohms.  (79)

By matching the reactive component (X_(in)) seen “looking down” into the lossy conducting medium 203 with the reactive component (X_(base)) seen “looking up” into the guided surface wave probe 200, the coupling into the guided surface waveguide mode may be maximized. This can be accomplished by adjusting the capacitance of the charge terminal T₁ without changing the traveling wave phase delays of the coil and vertical feed line conductor. For example, by adjusting the charge terminal capacitance (C_(T)) to 61.8126 pF, the load impedance from Equation (62) is:

$\begin{matrix} {{Z_{L} = {\frac{1}{j\; \omega \; C_{T}} = {{- j}\mspace{14mu} 1392\mspace{14mu} {ohms}}}},} & (80) \end{matrix}$

and the reactive components at the boundary are matched.

Using Equation (51), the impedance of the vertical feed line conductor (having a diameter (2a) of 0.27 inches) is given as

$\begin{matrix} {{Z_{w} = {{138\mspace{14mu} \log \mspace{14mu} \left( \frac{1.123\mspace{14mu} V_{w}\lambda_{0}}{2\; \pi \; a} \right)} = {537.534\mspace{14mu} {ohms}}}},} & (81) \end{matrix}$

and the impedance seen “looking up” into the vertical feed line conductor is given by Equation (63) as:

$\begin{matrix} {Z_{2} = {{Z_{W}\frac{z_{L} + {z_{W}{\tanh \left( {j\; \theta_{y}} \right)}}}{z_{W} + {z_{L}{\tanh \left( {j\; \theta_{y}} \right)}}}} = {{- j}\mspace{11mu} 835.438\mspace{14mu} {{ohms}.}}}} & (82) \end{matrix}$

Using Equation (47), the characteristic impedance of the helical coil is given as

$\begin{matrix} {{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\; {n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack} = {1446\mspace{14mu} {ohms}}}},} & (83) \end{matrix}$

and the impedance seen “looking up” into the coil at the base is given by Equation (64) as:

$\begin{matrix} {Z_{base} = {{Z_{c}\frac{z_{2} + {z_{c}{\tanh \left( {j\; \theta_{c}} \right)}}}{z_{c} + {z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}} = {{- j}\mspace{14mu} 26.271\mspace{14mu} {{ohms}.}}}} & (84) \end{matrix}$

When compared to the solution of Equation (79), it can be seen that the reactive components are opposite and approximately equal, and thus are conjugates of each other. Thus, the impedance (Z_(ip)) seen “looking up” into the equivalent image plane model of FIGS. 9A and 9B from the perfectly conducting image ground plane is only resistive or Z_(ip)=R+j0.

When the electric fields produced by a guided surface waveguide probe 200 (FIG. 3) are established by matching the traveling wave phase delay of the feed network to the wave tilt angle and the probe structure is resonated with respect to the perfectly conducting image ground plane at complex depth z=−d/2, the fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided surface traveling wave is launched along the surface of the lossy conducting medium. As illustrated in FIG. 1, the guided field strength curve 103 of the guided electromagnetic field has a characteristic exponential decay of e^(−αd)/√{square root over (d)} and exhibits a distinctive knee 109 on the log-log scale.

In summary, both analytically and experimentally, the traveling wave component on the structure of the guided surface waveguide probe 200 has a phase delay (Φ) at its upper terminal that matches the angle (Ψ) of the wave tilt of the surface traveling wave (Φ=Ψ). Under this condition, the surface waveguide may be considered to be “mode-matched”. Furthermore, the resonant standing wave component on the structure of the guided surface waveguide probe 200 has a V_(MAX) at the charge terminal T₁ and a V_(MIN) down at the image plane 139 (FIG. 8B) where Z_(ip)=R_(ip)+j 0 at a complex depth of z=−d/2, not at the connection at the physical boundary 136 of the lossy conducting medium 203 (FIG. 8B). Lastly, the charge terminal T₁ is of sufficient height H₁ of FIG. 3 (h≥R_(x) tan ψ_(i,B)) so that electromagnetic waves incident onto the lossy conducting medium 203 at the complex Brewster angle do so out at a distance (≥R_(x)) where the 1/√{square root over (r)} term is predominant. Receive circuits can be utilized with one or more guided surface waveguide probes to facilitate wireless transmission and/or power delivery systems.

Referring back to FIG. 3, operation of a guided surface waveguide probe 200 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, an adaptive probe control system 230 can be used to control the feed network 209 and/or the charge terminal T₁ to control the operation of the guided surface waveguide probe 200. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g., conductivity σ and relative permittivity ε_(r)), variations in field strength and/or variations in loading of the guided surface waveguide probe 200. As can be seen from Equations (31), (41) and (42), the index of refraction (n), the complex Brewster angle (θ_(i,B)), and the wave tilt (| W|e^(jΨ)) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the adaptive probe control system 230. The probe control system 230 can then make one or more adjustments to the guided surface waveguide probe 200 to maintain specified operational conditions for the guided surface waveguide probe 200. For instance, as the moisture and temperature vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 200. Generally, it would be desirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R_(x) for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe 200.

The conductivity measurement probes and/or permittivity sensors can be configured to evaluate the conductivity and/or permittivity on a periodic basis and communicate the information to the probe control system 230. The information may be communicated to the probe control system 230 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate wired or wireless communication network. Based upon the monitored conductivity and/or permittivity, the probe control system 230 may evaluate the variation in the index of refraction (n), the complex Brewster angle (θ_(i,B)), and/or the wave tilt (|W|e^(jΨ)) and adjust the guided surface waveguide probe 200 to maintain the phase delay (Φ) of the feed network 209 equal to the wave tilt angle (Ψ) and/or maintain resonance of the equivalent image plane model of the guided surface waveguide probe 200. This can be accomplished by adjusting, e.g., θ_(y), θ_(c) and/or C_(T). For instance, the probe control system 230 can adjust the self-capacitance of the charge terminal T₁ and/or the phase delay θ_(y), θ_(c)) applied to the charge terminal T₁ to maintain the electrical launching efficiency of the guided surface wave at or near its maximum. For example, the self-capacitance of the charge terminal T₁ can be varied by changing the size of the terminal. The charge distribution can also be improved by increasing the size of the charge terminal T₁, which can reduce the chance of an electrical discharge from the charge terminal T₁. In other embodiments, the charge terminal T₁ can include a variable inductance that can be adjusted to change the load impedance Z_(L). The phase applied to the charge terminal T₁ can be adjusted by varying the tap position on the coil 215 (FIG. 7), and/or by including a plurality of predefined taps along the coil 215 and switching between the different predefined tap locations to maximize the launching efficiency.

Field or field strength (FS) meters may also be distributed about the guided surface waveguide probe 200 to measure field strength of fields associated with the guided surface wave. The field or FS meters can be configured to detect the field strength and/or changes in the field strength (e.g., electric field strength) and communicate that information to the probe control system 230. The information may be communicated to the probe control system 230 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. As the load and/or environmental conditions change or vary during operation, the guided surface waveguide probe 200 may be adjusted to maintain specified field strength(s) at the FS meter locations to ensure appropriate power transmission to the receivers and the loads they supply.

For example, the phase delay (Φ=θ_(y)+θ_(c)) applied to the charge terminal T₁ can be adjusted to match the wave tilt angle (Ψ). By adjusting one or both phase delays, the guided surface waveguide probe 200 can be adjusted to ensure the wave tilt corresponds to the complex Brewster angle. This can be accomplished by adjusting a tap position on the coil 215 (FIG. 7) to change the phase delay supplied to the charge terminal T₁. The voltage level supplied to the charge terminal T₁ can also be increased or decreased to adjust the electric field strength. This may be accomplished by adjusting the output voltage of the excitation source 212 or by adjusting or reconfiguring the feed network 209. For instance, the position of the tap 227 (FIG. 7) for the AC source 212 can be adjusted to increase the voltage seen by the charge terminal Maintaining field strength levels within predefined ranges can improve coupling by the receivers, reduce ground current losses, and avoid interference with transmissions from other guided surface waveguide probes 200.

The probe control system 230 can be implemented with hardware, firmware, software executed by hardware, or a combination thereof. For example, the probe control system 230 can include processing circuitry including a processor and a memory, both of which can be coupled to a local interface such as, for example, a data bus with an accompanying control/address bus as can be appreciated by those with ordinary skill in the art. A probe control application may be executed by the processor to adjust the operation of the guided surface waveguide probe 200 based upon monitored conditions. The probe control system 230 can also include one or more network interfaces for communicating with the various monitoring devices. Communications can be through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. The probe control system 230 may comprise, for example, a computer system such as a server, desktop computer, laptop, or other system with like capability.

Referring back to the example of FIG. 5A, the complex angle trigonometry is shown for the ray optic interpretation of the incident electric field (E) of the charge terminal T₁ with a complex Brewster angle (θ_(i,B)) at the Hankel crossover distance (R_(x)). Recall that, for a lossy conducting medium, the Brewster angle is complex and specified by equation (38). Electrically, the geometric parameters are related by the electrical effective height (h_(eff)) of the charge terminal T₁ by equation (39). Since both the physical height (h_(p)) and the Hankel crossover distance (R_(x)) are real quantities, the angle of the desired guided surface wave tilt at the Hankel crossover distance (W_(Rx)) is equal to the phase (Φ) of the complex effective height (h_(eff)). With the charge terminal T₁ positioned at the physical height h_(p) and excited with a charge having the appropriate phase Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R_(x), and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection.

However, Equation (39) means that the physical height of the guided surface waveguide probe 200 can be relatively small. While this will excite the guided surface waveguide mode, this can result in an unduly large bound charge with little free charge. To compensate, the charge terminal T₁ can be raised to an appropriate elevation to increase the amount of free charge. As one example rule of thumb, the charge terminal T₁ can be positioned at an elevation of about 4-5 times (or more) the effective diameter of the charge terminal T₁. FIG. 6 illustrates the effect of raising the charge terminal T₁ above the physical height (h_(p)) shown in FIG. 5A. The increased elevation causes the distance at which the wave tilt is incident with the lossy conductive medium to move beyond the Hankel crossover point 121 (FIG. 5A). To improve coupling in the guided surface waveguide mode, and thus provide for a greater launching efficiency of the guided surface wave, a lower compensation terminal T₂ can be used to adjust the total effective height (h_(TE)) of the charge terminal T₁ such that the wave tilt at the Hankel crossover distance is at the Brewster angle.

Referring to FIG. 12, shown is an example of a guided surface waveguide probe 200 c that includes an elevated charge terminal T₁ and a lower compensation terminal T₂ that are arranged along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. In this respect, the charge terminal T₁ is placed directly above the compensation terminal T₂ although it is possible that some other arrangement of two or more charge and/or compensation terminals TN can be used. The guided surface waveguide probe 200 c is disposed above a lossy conducting medium 203 according to an embodiment of the present disclosure. The lossy conducting medium 203 makes up Region 1 with a second medium 206 that makes up Region 2 sharing a boundary interface with the lossy conducting medium 203.

The guided surface waveguide probe 200 c includes a coupling circuit 209 that couples an excitation source 212 to the charge terminal T₁ and the compensation terminal T₂. According to various embodiments, charges Q₁ and Q₂ can be imposed on the respective charge and compensation terminals T₁ and T₂, depending on the voltages applied to terminals T₁ and T₂ at any given instant. I₁ is the conduction current feeding the charge Q₁ on the charge terminal T₁ via the terminal lead, and I₂ is the conduction current feeding the charge Q₂ on the compensation terminal T₂ via the terminal lead.

According to the embodiment of FIG. 12, the charge terminal T₁ is positioned over the lossy conducting medium 203 at a physical height H₁, and the compensation terminal T₂ is positioned directly below T₁ along the vertical axis z at a physical height H₂, where H₂ is less than H₁. The height h of the transmission structure may be calculated as h=H₁−H₂. The charge terminal T₁ has an isolated (or self) capacitance C₁, and the compensation terminal T₂ has an isolated (or self) capacitance C₂. A mutual capacitance CM can also exist between the terminals T₁ and T₂ depending on the distance therebetween. During operation, charges Q₁ and Q₂ are imposed on the charge terminal T₁ and the compensation terminal T₂, respectively, depending on the voltages applied to the charge terminal T₁ and the compensation terminal T₂ at any given instant.

Referring next to FIG. 13, shown is a ray optics interpretation of the effects produced by the elevated charge Q₁ on charge terminal T₁ and compensation terminal T₂ of FIG. 12. With the charge terminal T₁ elevated to a height where the ray intersects with the lossy conductive medium at the Brewster angle at a distance greater than the Hankel crossover point 121 as illustrated by line 163, the compensation terminal T₂ can be used to adjust h_(TE) by compensating for the increased height. The effect of the compensation terminal T₂ is to reduce the electrical effective height of the guided surface waveguide probe (or effectively raise the lossy medium interface) such that the wave tilt at the Hankel crossover distance is at the Brewster angle as illustrated by line 166.

The total effective height can be written as the superposition of an upper effective height (h_(UE)) associated with the charge terminal T₁ and a lower effective height (h_(LE)) associated with the compensation terminal T₂ such that

h _(TE) =h _(UE) h _(LE) =h _(p) e ^(j(βh) ^(p) ^(+Φ) ^(U) ⁾ +h _(d) e ^(j(βh) ^(d) ^(+Φ) ^(L) ⁾ =R _(x) ×W,  (85)

where Φ_(U) is the phase delay applied to the upper charge terminal T₁, Φ_(L) is the phase delay applied to the lower compensation terminal T₂, β=2π/λ_(p) is the propagation factor from Equation (35), h_(p) is the physical height of the charge terminal T₁ and h_(d) is the physical height of the compensation terminal T₂. If extra lead lengths are taken into consideration, they can be accounted for by adding the charge terminal lead length z to the physical height h_(p) of the charge terminal T₁ and the compensation terminal lead length y to the physical height h_(d) of the compensation terminal T₂ as shown in

h _(TE)=(h _(p) +z)e ^(j(β(h) ^(p) ^(+z)+Φ) ^(U) ⁾+(h _(d) +y)e ^(j(β(h) ^(d) ^(+y)+Φ) ^(L) ⁾ =R _(x) ×W.  (86)

The lower effective height can be used to adjust the total effective height (h_(TE)) to equal the complex effective height (h_(eff)) of FIG. 5A.

Equations (85) or (86) can be used to determine the physical height of the lower disk of the compensation terminal T₂ and the phase angles to feed the terminals in order to obtain the desired wave tilt at the Hankel crossover distance. For example, Equation (86) can be rewritten as the phase shift applied to the charge terminal T₁ as a function of the compensation terminal height (h_(d)) to give

$\begin{matrix} {{\Phi_{U}\left( h_{d} \right)} = {{- {\beta \left( {h_{p} + z} \right)}} - {j\mspace{11mu} {{\ln \left( \frac{{R_{x} \times W} - {\left( {h_{d} + y} \right)e^{j{({{\beta \; h_{d}} + {\beta \; y} + \Phi_{L}})}}}}{\left( {h_{p} + z} \right)} \right)}.}}}} & (87) \end{matrix}$

To determine the positioning of the compensation terminal T₂, the relationships discussed above can be utilized. First, the total effective height (h_(TE)) is the superposition of the complex effective height (h_(UE)) of the upper charge terminal T₁ and the complex effective height (h_(LE)) of the lower compensation terminal T₂ as expressed in Equation (86). Next, the tangent of the angle of incidence can be expressed geometrically as

$\begin{matrix} {{{\tan \; \psi_{E}} = \frac{h_{TE}}{R_{x}}},} & (88) \end{matrix}$

which is equal to the definition of the wave tilt, W. Finally, given the desired Hankel crossover distance R_(x), the h_(TE) can be adjusted to make the wave tilt of the incident ray match the complex Brewster angle at the Hankel crossover point 121. This can be accomplished by adjusting h_(p), Φ_(U), and/or h_(d).

These concepts may be better understood when discussed in the context of an example of a guided surface waveguide probe. Referring to FIG. 14, shown is a graphical representation of an example of a guided surface waveguide probe 200 d including an upper charge terminal T₁ (e.g., a sphere at height h_(T)) and a lower compensation terminal T₂ (e.g., a disk at height h_(d)) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. During operation, charges Q₁ and Q₂ are imposed on the charge and compensation terminals T₁ and T₂, respectively, depending on the voltages applied to the terminals T₁ and T₂ at any given instant.

An AC source 212 acts as the excitation source for the charge terminal which is coupled to the guided surface waveguide probe 200 d through a coupling circuit 209 comprising a coil 215 such as, e.g., a helical coil. The AC source 212 can be connected across a lower portion of the coil 215 through a tap 227, as shown in FIG. 14, or can be inductively coupled to the coil 215 by way of a primary coil. The coil 215 can be coupled to a ground stake 218 at a first end and the charge terminal T₁ at a second end. In some implementations, the connection to the charge terminal T₁ can be adjusted using a tap 224 at the second end of the coil 215. The compensation terminal T₂ is positioned above and substantially parallel with the lossy conducting medium 203 (e.g., the ground or Earth), and energized through a tap 233 coupled to the coil 215. An ammeter 236 located between the coil 215 and ground stake 218 can be used to provide an indication of the magnitude of the current flow (I₀) at the base of the guided surface waveguide probe. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 218 to obtain an indication of the magnitude of the current flow (I₀).

In the example of FIG. 14, the coil 215 is coupled to a ground stake 218 at a first end and the charge terminal T₁ at a second end via a vertical feed line conductor 221. In some implementations, the connection to the charge terminal T₁ can be adjusted using a tap 224 at the second end of the coil 215 as shown in FIG. 14. The coil 215 can be energized at an operating frequency by the AC source 212 through a tap 227 at a lower portion of the coil 215. In other implementations, the AC source 212 can be inductively coupled to the coil 215 through a primary coil. The compensation terminal T₂ is energized through a tap 233 coupled to the coil 215. An ammeter 236 located between the coil 215 and ground stake 218 can be used to provide an indication of the magnitude of the current flow at the base of the guided surface waveguide probe 200 d. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 218 to obtain an indication of the magnitude of the current flow. The compensation terminal T₂ is positioned above and substantially parallel with the lossy conducting medium 203 (e.g., the ground).

In the example of FIG. 14, the connection to the charge terminal T₁ located on the coil 215 above the connection point of tap 233 for the compensation terminal T₂. Such an adjustment allows an increased voltage (and thus a higher charge Q₁) to be applied to the upper charge terminal T₁. In other embodiments, the connection points for the charge terminal T₁ and the compensation terminal T₂ can be reversed. It is possible to adjust the total effective height (h_(TE)) of the guided surface waveguide probe 200 d to excite an electric field having a guided surface wave tilt at the Hankel crossover distance R_(x). The Hankel crossover distance can also be found by equating the magnitudes of equations (20b) and (21) for −jγρ, and solving for R_(x) as illustrated by FIG. 4. The index of refraction (n), the complex Brewster angle (θ_(i,B) and ψ_(i,B)), the wave tilt (|W|e^(jΨ)) and the complex effective height (h_(eff)=h_(p)e^(jΦ)) can be determined as described with respect to Equations (41)-(44) above.

With the selected charge terminal T₁ configuration, a spherical diameter (or the effective spherical diameter) can be determined. For example, if the charge terminal T₁ is not configured as a sphere, then the terminal configuration may be modeled as a spherical capacitance having an effective spherical diameter. The size of the charge terminal T₁ can be chosen to provide a sufficiently large surface for the charge Q₁ imposed on the terminals. In general, it is desirable to make the charge terminal T₁ as large as practical. The size of the charge terminal T₁ should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. To reduce the amount of bound charge on the charge terminal T₁, the desired elevation to provide free charge on the charge terminal T₁ for launching a guided surface wave should be at least 4-5 times the effective spherical diameter above the lossy conductive medium (e.g., the Earth). The compensation terminal T₂ can be used to adjust the total effective height (h_(TE)) of the guided surface waveguide probe 200 d to excite an electric field having a guided surface wave tilt at R_(x). The compensation terminal T₂ can be positioned below the charge terminal T₁ at h_(d)=h_(T)−h_(p), where h_(T) is the total physical height of the charge terminal T₁. With the position of the compensation terminal T₂ fixed and the phase delay Φ_(U) applied to the upper charge terminal T₁, the phase delay Φ_(L) applied to the lower compensation terminal T₂ can be determined using the relationships of Equation (86), such that:

$\begin{matrix} {{\Phi_{U}\left( h_{d} \right)} = {{- {\beta \left( {h_{p} + y} \right)}} - {j\mspace{11mu} {{\ln \left( \frac{{R_{x} \times W} - {\left( {h_{p} + z} \right)e^{j{({{\beta \; h_{p}} + {\beta \; z} + \Phi_{L}})}}}}{\left( {h_{d} + y} \right)} \right)}.}}}} & (89) \end{matrix}$

In alternative embodiments, the compensation terminal T₂ can be positioned at a height h_(d) where Im{Φ_(L)}=0. This is graphically illustrated in FIG. 15A, which shows plots 172 and 175 of the imaginary and real parts of Φ_(U), respectively. The compensation terminal T₂ is positioned at a height h_(d) where Im{Φ_(U)}=0, as graphically illustrated in plot 172. At this fixed height, the coil phase Φ_(U) can be determined from Re{Φ_(U)}, as graphically illustrated in plot 175.

With the AC source 212 coupled to the coil 215 (e.g., at the 50Ω point to maximize coupling), the position of tap 233 may be adjusted for parallel resonance of the compensation terminal T₂ with at least a portion of the coil at the frequency of operation. FIG. 15B shows a schematic diagram of the general electrical hookup of FIG. 14 in which V₁ is the voltage applied to the lower portion of the coil 215 from the AC source 212 through tap 227, V₂ is the voltage at tap 224 that is supplied to the upper charge terminal T₁, and V₃ is the voltage applied to the lower compensation terminal T₂ through tap 233. The resistances R_(p) and R_(d) represent the ground return resistances of the charge terminal T₁ and compensation terminal T₂, respectively. The charge and compensation terminals T₁ and T₂ may be configured as spheres, cylinders, toroids, rings, hoods, or any other combination of capacitive structures. The size of the charge and compensation terminals T₁ and T₂ can be chosen to provide a sufficiently large surface for the charges Q₁ and Q₂ imposed on the terminals. In general, it is desirable to make the charge terminal T₁ as large as practical. The size of the charge terminal T₁ should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. The self-capacitance C_(p) and C_(d) of the charge and compensation terminals T₁ and T₂ respectively, can be determined using, for example, equation (24).

As can be seen in FIG. 15B, a resonant circuit is formed by at least a portion of the inductance of the coil 215, the self-capacitance C_(d) of the compensation terminal T₂, and the ground return resistance R_(d) associated with the compensation terminal T₂. The parallel resonance can be established by adjusting the voltage V₃ applied to the compensation terminal T₂ (e.g., by adjusting a tap 233 position on the coil 215) or by adjusting the height and/or size of the compensation terminal T₂ to adjust C_(d). The position of the coil tap 233 can be adjusted for parallel resonance, which will result in the ground current through the ground stake 218 and through the ammeter 236 reaching a maximum point. After parallel resonance of the compensation terminal T₂ has been established, the position of the tap 227 for the AC source 212 can be adjusted to the 50Ω point on the coil 215.

Voltage V₂ from the coil 215 can be applied to the charge terminal T₁, and the position of tap 224 can be adjusted such that the phase (Φ) of the total effective height (h_(TE)) approximately equals the angle of the guided surface wave tilt (W_(Rx)) at the Hankel crossover distance (R_(x)). The position of the coil tap 224 can be adjusted until this operating point is reached, which results in the ground current through the ammeter 236 increasing to a maximum. At this point, the resultant fields excited by the guided surface waveguide probe 200 d are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 203, resulting in the launching of a guided surface wave along the surface of the lossy conducting medium 203. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 200.

Resonance of the circuit including the compensation terminal T₂ may change with the attachment of the charge terminal T₁ and/or with adjustment of the voltage applied to the charge terminal T₁ through tap 224. While adjusting the compensation terminal circuit for resonance aids the subsequent adjustment of the charge terminal connection, it is not necessary to establish the guided surface wave tilt (W_(Rx)) at the Hankel crossover distance (R_(x)). The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap 227 for the AC source 212 to be at the 50Ω point on the coil 215 and adjusting the position of tap 233 to maximize the ground current through the ammeter 236. Resonance of the circuit including the compensation terminal T₂ may drift as the positions of taps 227 and 233 are adjusted, or when other components are attached to the coil 215.

In other implementations, the voltage V₂ from the coil 215 can be applied to the charge terminal T₁, and the position of tap 233 can be adjusted such that the phase (Φ) of the total effective height (h_(TE)) approximately equals the angle (Ψ) of the guided surface wave tilt at R_(x). The position of the coil tap 224 can be adjusted until the operating point is reached, resulting in the ground current through the ammeter 236 substantially reaching a maximum. The resultant fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 203, and a guided surface wave is launched along the surface of the lossy conducting medium 203. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 200. The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap 227 for the AC source 212 to be at the 50Ω point on the coil 215 and adjusting the position of tap 224 and/or 233 to maximize the ground current through the ammeter 236.

Referring back to FIG. 12, operation of a guided surface waveguide probe 200 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, a probe control system 230 can be used to control the coupling circuit 209 and/or positioning of the charge terminal T₁ and/or compensation terminal T₂ to control the operation of the guided surface waveguide probe 200. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g., conductivity a and relative permittivity ε_(r)), variations in field strength and/or variations in loading of the guided surface waveguide probe 200. As can be seen from Equations (41)-(44), the index of refraction (n), the complex Brewster angle (θ_(i,B) and ψ_(i,B)), the wave tilt (|W|e^(jΨ)) and the complex effective height (h_(eff)=h_(p)e^(jΦ) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the probe control system 230. The probe control system 230 can then make one or more adjustments to the guided surface waveguide probe 200 to maintain specified operational conditions for the guided surface waveguide probe 200. For instance, as the moisture and temperature vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 200. Generally, it would be desirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R_(x) for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe 200.

With reference then to FIG. 16, shown is an example of a guided surface waveguide probe 200 e that includes a charge terminal T₁ and a charge terminal T₂ that are arranged along a vertical axis z. The guided surface waveguide probe 200 e is disposed above a lossy conducting medium 203, which makes up Region 1. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Region 2. The charge terminals T₁ and T₂ are positioned over the lossy conducting medium 203. The charge terminal T₁ is positioned at height H₁, and the charge terminal T₂ is positioned directly below T₁ along the vertical axis z at height H₂, where H₂ is less than H₁. The height h of the transmission structure presented by the guided surface waveguide probe 200 e is h=H₁−H₂. The guided surface waveguide probe 200 e includes a probe coupling circuit 209 that couples an excitation source 212 to the charge terminals T₁ and T₂.

The charge terminals T₁ and/or T₂ include a conductive mass that can hold an electrical charge, which may be sized to hold as much charge as practically possible. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂, which can be determined using, for example, equation (24). By virtue of the placement of the charge terminal T₁ directly above the charge terminal T₂, a mutual capacitance C_(M) is created between the charge terminals T₁ and T₂. Note that the charge terminals T₁ and T₂ need not be identical, but each can have a separate size and shape, and can include different conducting materials. Ultimately, the field strength of a guided surface wave launched by a guided surface waveguide probe 200 e is directly proportional to the quantity of charge on the terminal T₁. The charge Q₁ is, in turn, proportional to the self-capacitance C₁ associated with the charge terminal T₁ since Q₁=C₁V, where V is the voltage imposed on the charge terminal T₁.

When properly adjusted to operate at a predefined operating frequency, the guided surface waveguide probe 200 e generates a guided surface wave along the surface of the lossy conducting medium 203. The excitation source 212 can generate electrical energy at the predefined frequency that is applied to the guided surface waveguide probe 200 e to excite the structure. When the electromagnetic fields generated by the guided surface waveguide probe 200 e are substantially mode-matched with the lossy conducting medium 203, the electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle that results in little or no reflection. Thus, the surface waveguide probe 200 e does not produce a radiated wave, but launches a guided surface traveling wave along the surface of a lossy conducting medium 203. The energy from the excitation source 212 can be transmitted as Zenneck surface currents to one or more receivers that are located within an effective transmission range of the guided surface waveguide probe 200 e.

One can determine asymptotes of the radial Zenneck surface current J_(ρ)(ρ) on the surface of the lossy conducting medium 203 to be J₁(ρ) close-in and J₂ (ρ) far-out, where

$\begin{matrix} {{{{{Close}\text{-}{in}\mspace{11mu} \left( {\rho < {\lambda \text{/}8}} \right)\text{:}\mspace{14mu} {J_{\rho}(\rho)}} \sim J_{1}} = {\frac{I_{1} + I_{2}}{2\pi \; \rho} + \frac{{E_{\rho}^{QS}\left( Q_{1} \right)} + {E_{\rho}^{QS}\left( Q_{2} \right)}}{z_{\rho}}}},\mspace{20mu} {and}} & (90) \\ {{{{Far}\text{-}{out}\mspace{11mu} \left( {\rho\operatorname{>>}{\lambda/8}} \right)\text{:}\mspace{14mu} {J_{\rho}(\rho)}} \sim J_{2}} = {\frac{j\; \gamma \; \omega \; Q_{1}}{4} \times \sqrt{\frac{2\gamma}{\pi}} \times {\frac{e^{{- {({\alpha + {j\; \beta}})}}\rho}}{\sqrt{\rho}}.}}} & (91) \end{matrix}$

where I₁ is the conduction current feeding the charge Q₁ on the first charge terminal and I₂ is the conduction current feeding the charge Q₂ on the second charge terminal T₂. The charge Q₁ on the upper charge terminal T₁ is determined by Q₁=C₁V₁, where C₁ is the isolated capacitance of the charge terminal T₁. Note that there is a third component to J₁ set forth above given by (E_(ρ) ^(Q) ¹ )/Z_(ρ), which follows from the Leontovich boundary condition and is the radial current contribution in the lossy conducting medium 203 pumped by the quasi-static field of the elevated oscillating charge on the first charge terminal Q₁. The quantity Z_(ρ)=jωμ_(o)/γ_(e) is the radial impedance of the lossy conducting medium, where γ_(e)=(jωμ₁σ₁−ω²μ₁ε₁)^(1/2).

The asymptotes representing the radial current close-in and far-out as set forth by equations (90) and (91) are complex quantities. According to various embodiments, a physical surface current J(ρ), is synthesized to match as close as possible the current asymptotes in magnitude and phase. That is to say close-in, |J(ρ)| is to be tangent to |J₁|, and far-out |J(ρ)| is to be tangent to |J₂|. Also, according to the various embodiments, the phase of J(ρ) should transition from the phase of J₁ close-in to the phase of J₂ far-out.

In order to match the guided surface wave mode at the site of transmission to launch a guided surface wave, the phase of the surface current |J₂| far-out should differ from the phase of the surface current |J₁| close-in by the propagation phase corresponding to e^(−jβ(ρ) ² ^(−ρ) ¹ ⁾ plus a constant of approximately 45 degrees or 225 degrees. This is because there are two roots for √{square root over (γ)}, one near π/4 and one near 5π/4. The properly adjusted synthetic radial surface current is

$\begin{matrix} {{J_{\rho}\left( {\rho,\varphi,0} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}.}}} & (92) \end{matrix}$

Note that this is consistent with equation (17). By Maxwell's equations, such a J(ρ) surface current automatically creates fields that conform to

$\begin{matrix} {{H_{\varphi} = {\frac{{- \gamma}\; I_{o}}{4}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},} & (93) \\ {{E_{\rho} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\; \omega \; ɛ_{o}} \right)e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},{and}} & (94) \\ {E_{z} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{- \gamma}{\omega \; ɛ_{o}} \right)e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}.}}} & (95) \end{matrix}$

Thus, the difference in phase between the surface current |J₂| far-out and the surface current |J₁| close-in for the guided surface wave mode that is to be matched is due to the characteristics of the Hankel functions in equations (93)-(95), which are consistent with equations (1)-(3). It is of significance to recognize that the fields expressed by equations (1)-(6) and (17) and equations (92)-(95) have the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation.

In order to obtain the appropriate voltage magnitudes and phases for a given design of a guided surface waveguide probe 200 e at a given location, an iterative approach may be used. Specifically, analysis may be performed of a given excitation and configuration of a guided surface waveguide probe 200 e taking into account the feed currents to the terminals T₁ and T₂, the charges on the charge terminals T₁ and T₂, and their images in the lossy conducting medium 203 in order to determine the radial surface current density generated. This process may be performed iteratively until an optimal configuration and excitation for a given guided surface waveguide probe 200 e is determined based on desired parameters. To aid in determining whether a given guided surface waveguide probe 200 e is operating at an optimal level, a guided field strength curve 103 (FIG. 1) may be generated using equations (1)-(12) based on values for the conductivity of Region 1 (σ₁) and the permittivity of Region 1 (ε₁) at the location of the guided surface waveguide probe 200 e. Such a guided field strength curve 103 can provide a benchmark for operation such that measured field strengths can be compared with the magnitudes indicated by the guided field strength curve 103 to determine if optimal transmission has been achieved.

In order to arrive at an optimized condition, various parameters associated with the guided surface waveguide probe 200 e may be adjusted. One parameter that may be varied to adjust the guided surface waveguide probe 200 e is the height of one or both of the charge terminals T₁ and/or T₂ relative to the surface of the lossy conducting medium 203. In addition, the distance or spacing between the charge terminals T₁ and T₂ may also be adjusted. In doing so, one may minimize or otherwise alter the mutual capacitance C_(M) or any bound capacitances between the charge terminals T₁ and T₂ and the lossy conducting medium 203 as can be appreciated. The size of the respective charge terminals T₁ and/or T₂ can also be adjusted. By changing the size of the charge terminals T₁ and/or T₂, one will alter the respective self-capacitances C₁ and/or C₂, and the mutual capacitance C_(M) as can be appreciated.

Still further, another parameter that can be adjusted is the probe coupling circuit 209 associated with the guided surface waveguide probe 200 e. This may be accomplished by adjusting the size of the inductive and/or capacitive reactances that make up the probe coupling circuit 209. For example, where such inductive reactances comprise coils, the number of turns on such coils may be adjusted. Ultimately, the adjustments to the probe coupling circuit 209 can be made to alter the electrical length of the probe coupling circuit 209, thereby affecting the voltage magnitudes and phases on the charge terminals T₁ and T₂.

Note that the iterations of transmission performed by making the various adjustments may be implemented by using computer models or by adjusting physical structures as can be appreciated. By making the above adjustments, one can create corresponding “close-in” surface current J₁ and “far-out” surface current J₂ that approximate the same currents J(ρ) of the guided surface wave mode specified in Equations (90) and (91) set forth above. In doing so, the resulting electromagnetic fields would be substantially or approximately mode-matched to a guided surface wave mode on the surface of the lossy conducting medium 203.

While not shown in the example of FIG. 16, operation of the guided surface waveguide probe 200 e may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, a probe control system 230 shown in FIG. 12 can be used to control the coupling circuit 209 and/or positioning and/or size of the charge terminals T₁ and/or T₂ to control the operation of the guided surface waveguide probe 200 e. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g., conductivity a and relative permittivity ε_(r)), variations in field strength and/or variations in loading of the guided surface waveguide probe 200 e.

Referring now to FIG. 17, shown is an example of the guided surface waveguide probe 200 e of FIG. 16, denoted herein as guided surface waveguide probe 200 f. The guided surface waveguide probe 200 f includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203 (e.g., the Earth). The second medium 206 is above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The guided surface waveguide probe 200 f includes a probe coupling circuit 209 that comprises an inductive impedance comprising a coil L_(1a) having a pair of leads that are coupled to respective ones of the charge terminals T₁ and T₂. In one embodiment, the coil L_(1a) is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the guided surface waveguide probe 200 f.

While the electrical length of the coil L_(1a) is specified as approximately one-half (½) the wavelength at the operating frequency, it is understood that the coil L_(1a) may be specified with an electrical length at other values. According to one embodiment, the fact that the coil L_(1a) has an electrical length of approximately one-half the wavelength at the operating frequency provides for an advantage in that a maximum voltage differential is created on the charge terminals T₁ and T₂. Nonetheless, the length or diameter of the coil L_(1a) may be increased or decreased when adjusting the guided surface waveguide probe 200 f to obtain optimal excitation of a guided surface wave mode. Adjustment of the coil length may be provided by taps located at one or both ends of the coil. In other embodiments, it may be the case that the inductive impedance is specified to have an electrical length that is significantly less than or greater than ½ the wavelength at the operating frequency of the guided surface waveguide probe 200 f.

The excitation source 212 can be coupled to the probe coupling circuit 209 by way of magnetic coupling. Specifically, the excitation source 212 is coupled to a coil L_(P) that is inductively coupled to the coil L_(1a). This may be done by link coupling, a tapped coil, a variable reactance, or other coupling approach as can be appreciated. To this end, the coil L_(P) acts as a primary, and the coil L_(1a) acts as a secondary as can be appreciated.

In order to adjust the guided surface waveguide probe 200 f for the transmission of a desired guided surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of the coil L_(1a) may be altered by adding or eliminating turns or by changing some other dimension of the coil L_(1a). The coil L_(1a) can also include one or more taps for adjusting the electrical length as shown in FIG. 17. The position of a tap connected to either charge terminal T₁ or T₂ can also be adjusted.

Referring next to FIGS. 18A, 18B, 18C and 19, shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery systems. FIGS. 18A and 18B-18C include a linear probe 303 and a tuned resonator 306, respectively. FIG. 19 is a magnetic coil 309 according to various embodiments of the present disclosure. According to various embodiments, each one of the linear probe 303, the tuned resonator 306, and the magnetic coil 309 may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 203 according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium 203 comprises a terrestrial medium (or Earth).

With specific reference to FIG. 18A, the open-circuit terminal voltage at the output terminals 312 of the linear probe 303 depends upon the effective height of the linear probe 303. To this end, the terminal point voltage may be calculated as

V _(T)=∫₀ ^(h) ^(e) E _(inc) ·dl,  (96)

where ε_(inc) is the strength of the incident electric field induced on the linear probe 303 in Volts per meter, dl is an element of integration along the direction of the linear probe 303, and h_(e) is the effective height of the linear probe 303. An electrical load 315 is coupled to the output terminals 312 through an impedance matching network 318.

When the linear probe 303 is subjected to a guided surface wave as described above, a voltage is developed across the output terminals 312 that may be applied to the electrical load 315 through a conjugate impedance matching network 318 as the case may be. In order to facilitate the flow of power to the electrical load 315, the electrical load 315 should be substantially impedance matched to the linear probe 303 as will be described below.

Referring to FIG. 18B, a ground current excited coil 306 a possessing a phase shift equal to the wave tilt of the guided surface wave includes a charge terminal T_(R) that is elevated (or suspended) above the lossy conducting medium 203. The charge terminal T_(R) has a self-capacitance C_(R). In addition, there may also be a bound capacitance (not shown) between the charge terminal T_(R) and the lossy conducting medium 203 depending on the height of the charge terminal T_(R) above the lossy conducting medium 203. The bound capacitance should preferably be minimized as much as is practicable, although this may not be entirely necessary in every instance.

The tuned resonator 306 a also includes a receiver network comprising a coil L_(R) having a phase shift Φ. One end of the coil L_(R) is coupled to the charge terminal T_(R), and the other end of the coil L_(R) is coupled to the lossy conducting medium 203. The receiver network can include a vertical supply line conductor that couples the coil L_(R) to the charge terminal T_(R). To this end, the coil L_(R) (which may also be referred to as tuned resonator L_(R)−C_(R)) comprises a series-adjusted resonator as the charge terminal C_(R) and the coil L_(R) are situated in series. The phase delay of the coil L_(R) can be adjusted by changing the size and/or height of the charge terminal T_(R), and/or adjusting the size of the coil L_(R) so that the phase Φ of the structure is made substantially equal to the angle of the wave tilt Ψ. The phase delay of the vertical supply line can also be adjusted by, e.g., changing length of the conductor.

For example, the reactance presented by the self-capacitance C_(R) is calculated as 1/jωC_(R). Note that the total capacitance of the structure 306 a may also include capacitance between the charge terminal T_(R) and the lossy conducting medium 203, where the total capacitance of the structure 306 a may be calculated from both the self-capacitance C_(R) and any bound capacitance as can be appreciated. According to one embodiment, the charge terminal T_(R) may be raised to a height so as to substantially reduce or eliminate any bound capacitance. The existence of a bound capacitance may be determined from capacitance measurements between the charge terminal T_(R) and the lossy conducting medium 203 as previously discussed.

The inductive reactance presented by a discrete-element coil L_(R) may be calculated as jωL, where L is the lumped-element inductance of the coil L_(R). If the coil L_(R) is a distributed element, its equivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the structure 306 a, one would make adjustments so that the phase delay is equal to the wave tilt for the purpose of mode-matching to the surface waveguide at the frequency of operation. Under this condition, the receiving structure may be considered to be “mode-matched” with the surface waveguide. A transformer link around the structure and/or an impedance matching network 324 may be inserted between the probe and the electrical load 327 in order to couple power to the load. Inserting the impedance matching network 324 between the probe terminals 321 and the electrical load 327 can effect a conjugate-match condition for maximum power transfer to the electrical load 327.

When placed in the presence of surface currents at the operating frequencies power will be delivered from the surface guided wave to the electrical load 327. To this end, an electrical load 327 may be coupled to the structure 306 a by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling. The elements of the coupling network may be lumped components or distributed elements as can be appreciated.

In the embodiment shown in FIG. 18B, magnetic coupling is employed where a coil Ls is positioned as a secondary relative to the coil L_(R) that acts as a transformer primary. The coil Ls may be link-coupled to the coil L_(R) by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the receiving structure 306 a comprises a series-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator of the appropriate phase delay may also be used.

While a receiving structure immersed in an electromagnetic field may couple energy from the field, it can be appreciated that polarization-matched structures work best by maximizing the coupling, and conventional rules for probe-coupling to waveguide modes should be observed. For example, a TE₂₀ (transverse electric mode) waveguide probe may be optimal for extracting energy from a conventional waveguide excited in the TE₂₀ mode. Similarly, in these cases, a mode-matched and phase-matched receiving structure can be optimized for coupling power from a surface-guided wave. The guided surface wave excited by a guided surface waveguide probe 200 on the surface of the lossy conducting medium 203 can be considered a waveguide mode of an open waveguide. Excluding waveguide losses, the source energy can be completely recovered. Useful receiving structures may be E-field coupled, H-field coupled, or surface-current excited.

The receiving structure can be adjusted to increase or maximize coupling with the guided surface wave based upon the local characteristics of the lossy conducting medium 203 in the vicinity of the receiving structure. To accomplish this, the phase delay (Φ) of the receiving structure can be adjusted to match the angle (Ψ) of the wave tilt of the surface traveling wave at the receiving structure. If configured appropriately, the receiving structure may then be tuned for resonance with respect to the perfectly conducting image ground plane at complex depth z=−d/2.

For example, consider a receiving structure comprising the tuned resonator 306 a of FIG. 18B, including a coil L_(R) and a vertical supply line connected between the coil L_(R) and a charge terminal T_(R). With the charge terminal T_(R) positioned at a defined height above the lossy conducting medium 203, the total phase shift Φ of the coil L_(R) and vertical supply line can be matched with the angle (Ψ) of the wave tilt at the location of the tuned resonator 306 a. From Equation (22), it can be seen that the wave tilt asymptotically passes to

$\begin{matrix} {{W = {{{W}e^{j\; \Psi}} = {\frac{E_{\rho}}{E_{z}}\underset{\mspace{20mu} {\rho\rightarrow\infty}\mspace{20mu}}{\rightarrow}\frac{1}{\sqrt{ɛ_{r} - {j\frac{\sigma_{1}}{{\omega ɛ}_{o}}}}}}}},} & (97) \end{matrix}$

where ε_(r) comprises the relative permittivity and σ₁ is the conductivity of the lossy conducting medium 203 at the location of the receiving structure, ε_(o) is the permittivity of free space, and ω=2πf, where f is the frequency of excitation. Thus, the wave tilt angle (Ψ) can be determined from Equation (97).

The total phase shift (Φ=θ_(c)+θ_(y)) of the tuned resonator 306 a includes both the phase delay (θ_(c)) through the coil L_(R) and the phase delay of the vertical supply line (θ_(y)). The spatial phase delay along the conductor length l_(w) of the vertical supply line can be given by θ_(y)=μ_(w)l_(w), where β_(w) is the propagation phase constant for the vertical supply line conductor. The phase delay due to the coil (or helical delay line) is θ_(c)=β_(p)l_(C), with a physical length of l_(C) and a propagation factor of

$\begin{matrix} {{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{0}}}},} & (98) \end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is the wavelength at the supplied frequency, and λ_(p) is the propagation wavelength resulting from the velocity factor V_(f). One or both of the phase delays (θ_(c)+θ_(y)) can be adjusted to match the phase shift Φ to the angle (Ψ) of the wave tilt. For example, a tap position may be adjusted on the coil L_(R) of FIG. 18B to adjust the coil phase delay (θ_(c)) to match the total phase shift to the wave tilt angle (Φ=Ψ). For example, a portion of the coil can be bypassed by the tap connection as illustrated in FIG. 18B. The vertical supply line conductor can also be connected to the coil L_(R) via a tap, whose position on the coil may be adjusted to match the total phase shift to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 306 a has been adjusted, the impedance of the charge terminal T_(R) can then be adjusted to tune to resonance with respect to the perfectly conducting image ground plane at complex depth z=−d/2. This can be accomplished by adjusting the capacitance of the charge terminal T₁ without changing the traveling wave phase delays of the coil L_(R) and vertical supply line. The adjustments are similar to those described with respect to FIGS. 9A and 9B.

The impedance seen “looking down” into the lossy conducting medium 203 to the complex image plane is given by:

Z _(in) =R _(in) +jX _(in) =Z _(o) tan h(jβ _(o)(d/2)),  (99)

where β_(o)=ω√{square root over (μ_(o)ε_(o))}. For vertically polarized sources over the Earth, the depth of the complex image plane can be given by:

d/2≈1/√{square root over (jωμ ₁σ₁−ω²μ₁ε₁)},  (100)

where μ₁ is the permeability of the lossy conducting medium 203 and ε₁=ε_(r)ε_(o).

At the base of the tuned resonator 306 a, the impedance seen “looking up” into the receiving structure is Z_(↑)=Z_(base) as illustrated in FIG. 9A. With a terminal impedance of:

$\begin{matrix} {{Z_{R} = \frac{1}{j\; \omega \; C_{R}}},} & (101) \end{matrix}$

where C_(R) is the self-capacitance of the charge terminal T_(R), the impedance seen “looking up” into the vertical supply line conductor of the tuned resonator 306 a is given by:

$\begin{matrix} {{Z_{2} = {{Z_{W}\frac{z_{R} + {z_{w}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{z_{w} + {z_{R}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{z_{R} + {z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{z_{w} + {z_{R}{\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (102) \end{matrix}$

and the impedance seen “looking up” into the coil L_(R) of the tuned resonator 306 a is given by:

$\begin{matrix} {Z_{base} = {{R_{base} + {jX}_{base}} = {{Z_{R}\frac{z_{2} + {z_{R}{\tanh \left( {j\; \beta_{p}H} \right)}}}{z_{R} + {z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{z_{2} + {z_{R}{\tanh\left( {j\; \theta_{c}}\; \right)}}}{z_{R} + {z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}.}}}}} & (103) \end{matrix}$

By matching the reactive component (X_(in)) seen “looking down” into the lossy conducting medium 203 with the reactive component (X_(base)) seen “looking up” into the tuned resonator 306 a, the coupling into the guided surface waveguide mode may be maximized.

Referring next to FIG. 18C, shown is an example of a tuned resonator 306 b that does not include a charge terminal T_(R) at the top of the receiving structure. In this embodiment, the tuned resonator 306 b does not include a vertical supply line coupled between the coil L_(R) and the charge terminal T_(R). Thus, the total phase shift (Φ) of the tuned resonator 306 b includes only the phase delay (θ_(c)) through the coil L_(R). As with the tuned resonator 306 a of FIG. 18B, the coil phase delay θ_(c) can be adjusted to match the angle (Ψ) of the wave tilt determined from Equation (97), which results in Φ=Ψ. While power extraction is possible with the receiving structure coupled into the surface waveguide mode, it is difficult to adjust the receiving structure to maximize coupling with the guided surface wave without the variable reactive load provided by the charge terminal T_(R).

Referring to FIG. 18D, shown is a flow chart 180 illustrating an example of adjusting a receiving structure to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium 203. Beginning with 181, if the receiving structure includes a charge terminal T_(R) (e.g., of the tuned resonator 306 a of FIG. 18B), then the charge terminal T_(R) is positioned at a defined height above a lossy conducting medium 203 at 184. As the surface guided wave has been established by a guided surface waveguide probe 200, the physical height (h_(p)) of the charge terminal T_(R) may be below that of the effective height. The physical height may be selected to reduce or minimize the bound charge on the charge terminal T_(R) (e.g., four times the spherical diameter of the charge terminal). If the receiving structure does not include a charge terminal T_(R) (e.g., of the tuned resonator 306 b of FIG. 18C), then the flow proceeds to 187.

At 187, the electrical phase delay Φ of the receiving structure is matched to the complex wave tilt angle Ψ defined by the local characteristics of the lossy conducting medium 203. The phase delay (θ_(c)) of the helical coil and/or the phase delay (θ_(y)) of the vertical supply line can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). The angle (Ψ) of the wave tilt can be determined from Equation (86). The electrical phase Φ can then be matched to the angle of the wave tilt. For example, the electrical phase delay Φ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters of the coil L_(R) and/or the length (or height) of the vertical supply line conductor.

Next at 190, the load impedance of the charge terminal T_(R) can be tuned to resonate the equivalent image plane model of the tuned resonator 306 a. The depth (d/2) of the conducting image ground plane 139 (FIG. 9A) below the receiving structure can be determined using Equation (100) and the values of the lossy conducting medium 203 (e.g., the Earth) at the receiving structure, which can be locally measured. Using that complex depth, the phase shift (θ_(d)) between the image ground plane 139 and the physical boundary 136 (FIG. 9A) of the lossy conducting medium 203 can be determined using θ_(d)=β_(o)d/2. The impedance (Z_(in)) as seen “looking down” into the lossy conducting medium 203 can then be determined using Equation (99). This resonance relationship can be considered to maximize coupling with the guided surface waves.

Based upon the adjusted parameters of the coil L_(R) and the length of the vertical supply line conductor, the velocity factor, phase delay, and impedance of the coil L_(R) and vertical supply line can be determined. In addition, the self-capacitance (C_(R)) of the charge terminal T_(R) can be determined using, e.g., Equation (24). The propagation factor (β_(p)) of the coil L_(R) can be determined using Equation (98), and the propagation phase constant (β_(w)) for the vertical supply line can be determined using Equation (49). Using the self-capacitance and the determined values of the coil L_(R) and vertical supply line, the impedance (Z_(base)) of the tuned resonator 306 a as seen “looking up” into the coil L_(R) can be determined using Equations (101), (102), and (103).

The equivalent image plane model of FIG. 9A also applies to the tuned resonator 306 a of FIG. 18B. The tuned resonator 306 a can be tuned to resonance with respect to the complex image plane by adjusting the load impedance Z_(R) of the charge terminal T_(R) such that the reactance component X_(base) of Z_(base) cancels out the reactance component of X_(in) of Z_(in), or X_(base)+X_(in)=0. Thus, the impedance at the physical boundary 136 (FIG. 9A) “looking up” into the coil of the tuned resonator 306 a is the conjugate of the impedance at the physical boundary 136 “looking down” into the lossy conducting medium 203. The load impedance Z_(R) can be adjusted by varying the capacitance (C_(R)) of the charge terminal T_(R) without changing the electrical phase delay Φ=θ_(c)+θ_(y) seen by the charge terminal T_(R). An iterative approach may be taken to tune the load impedance Z_(R) for resonance of the equivalent image plane model with respect to the conducting image ground plane 139. In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth) can be improved and/or maximized.

Referring to FIG. 19, the magnetic coil 309 comprises a receive circuit that is coupled through an impedance matching network 333 to an electrical load 336. In order to facilitate reception and/or extraction of electrical power from a guided surface wave, the magnetic coil 309 may be positioned so that the magnetic flux of the guided surface wave, H_(φ), passes through the magnetic coil 309, thereby inducing a current in the magnetic coil 309 and producing a terminal point voltage at its output terminals 330. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by

=∫∫_(A) _(CS) μ_(r)μ_(o)

·{circumflex over (n)}dA  (104)

where

is the coupled magnetic flux, μ_(r), is the effective relative permeability of the core of the magnetic coil 309, μ_(o) is the permeability of free space,

is the incident magnetic field strength vector, {circumflex over (n)} is a unit vector normal to the cross-sectional area of the turns, and A_(CS) is the area enclosed by each loop. For an N-turn magnetic coil 309 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil 309, the open-circuit induced voltage appearing at the output terminals 330 of the magnetic coil 309 is

$\begin{matrix} {{V = {{{- N}\frac{d\; \mathcal{F}}{dt}} \approx {{- j}\; \omega \; \mu_{r}\mu_{0}{NHA}_{CS}}}},} & (105) \end{matrix}$

where the variables are defined above. The magnetic coil 309 may be tuned to the guided surface wave frequency either as a distributed resonator or with an external capacitor across its output terminals 330, as the case may be, and then impedance-matched to an external electrical load 336 through a conjugate impedance matching network 333.

Assuming that the resulting circuit presented by the magnetic coil 309 and the electrical load 336 are properly adjusted and conjugate impedance matched, via impedance matching network 333, then the current induced in the magnetic coil 309 may be employed to optimally power the electrical load 336. The receive circuit presented by the magnetic coil 309 provides an advantage in that it does not have to be physically connected to the ground.

With reference to FIGS. 18A, 18B, 18C and 19, the receive circuits presented by the linear probe 303, the mode-matched structure 306, and the magnetic coil 309 each facilitate receiving electrical power transmitted from any one of the embodiments of guided surface waveguide probes 200 described above. To this end, the energy received may be used to supply power to an electrical load 315/327/336 via a conjugate matching network as can be appreciated. This contrasts with the signals that may be received in a receiver that were transmitted in the form of a radiated electromagnetic field. Such signals have very low available power, and receivers of such signals do not load the transmitters.

It is also characteristic of the present guided surface waves generated using the guided surface waveguide probes 200 described above that the receive circuits presented by the linear probe 303, the mode-matched structure 306, and the magnetic coil 309 will load the excitation source 212 (e.g., FIGS. 3, 12 and 16) that is applied to the guided surface waveguide probe 200, thereby generating the guided surface wave to which such receive circuits are subjected. This reflects the fact that the guided surface wave generated by a given guided surface waveguide probe 200 described above comprises a transmission line mode. By way of contrast, a power source that drives a radiating antenna that generates a radiated electromagnetic wave is not loaded by the receivers, regardless of the number of receivers employed.

Thus, together one or more guided surface waveguide probes 200 and one or more receive circuits in the form of the linear probe 303, the tuned mode-matched structure 306, and/or the magnetic coil 309 can make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a guided surface waveguide probe 200 as set forth above depends upon the frequency, it is possible that wireless power distribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution systems extensively investigated today include “energy harvesting” from radiation fields and also sensor coupling to inductive or reactive near-fields. In contrast, the present wireless-power system does not waste power in the form of radiation which, if not intercepted, is lost forever. Nor is the presently disclosed wireless-power system limited to extremely short ranges as with conventional mutual-reactance coupled near-field systems. The wireless-power system disclosed herein probe-couples to the novel surface-guided transmission line mode, which is equivalent to delivering power to a load by a waveguide or a load directly wired to the distant power generator. Not counting the power required to maintain transmission field strength plus that dissipated in the surface waveguide, which at extremely low frequencies is insignificant relative to the transmission losses in conventional high-tension power lines at 60 Hz, all of the generator power goes only to the desired electrical load. When the electrical load demand is terminated, the source power generation is relatively idle.

Referring next to FIGS. 20A-E, shown are examples of various schematic symbols that are used with reference to the discussion that follows. With specific reference to FIG. 20A, shown is a symbol that represents any one of the guided surface waveguide probes 200 a, 200 b, 200 c, 200 e, 200 d, or 200 f; or any variations thereof. In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface waveguide probe P. For the sake of simplicity in the following discussion, any reference to the guided surface waveguide probe P is a reference to any one of the guided surface waveguide probes 200 a, 200 b, 200 c, 200 e, 200 d, or 200 f; or variations thereof.

Similarly, with reference to FIG. 20B, shown is a symbol that represents a guided surface wave receive structure that may comprise any one of the linear probe 303 (e.g., FIG. 18A), the tuned resonator 306 (FIGS. 18B-18C), or the magnetic coil 309 (e.g., FIG. 19). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R. For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R is a reference to any one of the linear probe 303, the tuned resonator 306, or the magnetic coil 309; or variations thereof.

Further, with reference to FIG. 20C, shown is a symbol that specifically represents the linear probe 303 (FIG. 18A). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R_(P). For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R_(P) is a reference to the linear probe 303 or variations thereof.

Further, with reference to FIG. 20D, shown is a symbol that specifically represents the tuned resonator 306 (FIGS. 18B-18C). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R_(R). For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R_(R) is a reference to the tuned resonator 306 or variations thereof.

Further, with reference to FIG. 20E, shown is a symbol that specifically represents the magnetic coil 309 (FIG. 19). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R_(M). For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R_(M) is a reference to the magnetic coil 309 or variations thereof.

In many cases, the transmission of energy using guided surface waves requires relatively less infrastructure than other conventional means. For example, transmission lines are not necessary when using guided surface wave receive structures to receive electrical energy from a guided surface wave generated by a guided surface waveguide probe. Additionally, power generation may be centralized near a guided surface waveguide probe, while power consumption may be dispersed to a number of guided surface wave receive structures without, for example, transporting fuel to each location. Thus it is possible to quickly deploy power, for example, in emergency situations, over difficult terrain, or in other situations where setting up traditional power infrastructure is problematic.

Referring to FIG. 21, shown is an illustration depicting an overview of power distribution using a mobile guided surface wave transmit station 400, a mobile power repeater 409, and mobile guided surface wave receive stations 406 according to various embodiments of the present disclosure. The mobile guided surface wave receive stations 406 may comprise a portable kit as described in further detail below. In one embodiment, a mobile guided surface wave transmit station 400 can be located offshore, as shown in FIG. 21, and generate a first guided surface wave within a first transmission range 403. In other embodiments, the mobile guided surface wave transmit station 400 may be located at other offshore or onshore locations. Additionally, the transmission range 403 may be greater than that shown in FIG. 21 or even global in some embodiments.

A plurality of mobile guided surface wave receive stations 406 are shown located onshore in FIG. 21. A mobile power repeater 409 is also located onshore within the transmission range 403. The mobile power repeater 409 includes a guided surface wave receive structure and a guided surface waveguide probe. To this end, the mobile power repeater 409 may receive energy from a guided surface wave transmitted by the guided surface wave transmit station 400 offshore or at some other location. As shown in FIG. 21, the mobile power repeater 409 receives electrical energy from a guided surface wave transmitted by the mobile guided surface wave transmit station 400 on a ship at sea and generates a second guided surface wave at a desired transmission frequency within a second transmission range 412. The frequency of the second guided surface wave may differ from the frequency of the guided surface wave received by the mobile power repeater 409.

Some of the mobile guided surface wave receive stations 406 are within the transmission range 403, while others are outside the transmission range 403 but within the transmission range 412. Each of the mobile guided surface wave receive stations 406 includes a guided surface wave receive structure and can receive electrical energy from the first guided surface wave and/or the second guided surface wave to provide power to an electrical load at its location. Two of the mobile guided surface wave receive stations 406 are shown to be interconnected to feed a load 415. In this example, if each of the plurality mobile guided surface wave receive stations 406 is rated to receive 2 kVA, for example, and the load 415 requires 3 kVA, then a single mobile guided surface wave receive station 406 may be insufficient to power the load 415. On the other hand, two interconnected guided surface wave receive stations 406 may provide sufficient power. In this way, power is quickly distributable to all locations by way of the mobile guided surface wave receive stations 406 using mobile equipment and a relatively minimal amount of infrastructure.

Referring next to FIG. 22A, shown is a diagram that illustrates an example of a mobile guided surface wave transmit station 400 according to various embodiments of the present disclosure. The mobile guided surface wave transmit station 400 includes an excitation source 418, a matching network 421, and a guided surface waveguide probe 422. As shown, the excitation source 418 is connected to the matching network 421. The excitation source 418 may comprise any suitable power source such as a generator, solar panels, or batteries with an output converted to alternating current, or other power source. The matching network 421 is connected to the guided surface waveguide probe 422. In some embodiments, the matching network 421 is not necessary, such as where a connection is made with a tap on a coil (see FIG. 7). According to one embodiment, these components are mounted on a rigid frame for transport, according to the various embodiments.

The excitation source 418 generates power that is applied to the guided surface waveguide probe 422 through the matching network 421 (if needed). Consistent with the concepts described herein, the matching network 421 will, for example, seek to maximize power by effecting a conjugate match condition, and feed power to the guided surface waveguide probe 422 to generate a guided surface wave travelling along a terrestrial medium according to various embodiments.

Referring now to FIG. 22B, shown is a diagram that illustrates an example of a mobile power repeater 409 according to the various embodiments of the present disclosure. The mobile power repeater 409 includes a guided surface wave receive structure 410 that is connected to an AC-to-AC converter 411. An output of the AC-to-AC converter 411 is connected to a guided surface waveguide probe 422. In addition, although not shown, appropriate matching networks may be used between the guided surface wave receive structure 410 and the AC-to-AC converter 411 to provide higher power transfer as can be appreciated. According to one embodiment, these components are mounted on a rigid frame for transport.

In the embodiment shown in FIG. 22B, the power for the guided surface waveguide probe 421 comes from the guided surface wave receive structure 410, rather than the excitation source 418 of FIG. 22A. In this configuration, electrical energy may be received by the guided surface wave receive structure 410 at a first frequency and then retransmitted by the guided surface waveguide probe 421 as a guided surface wave along a terrestrial medium at a second frequency as desired at a given location. According to one embodiment, the frequency of transmission may be selectable.

Referring to FIG. 23, shown is an illustration of an example mobile guided surface wave transmit station 400 according to various embodiments of the present disclosure. In this embodiment, a fuel storage tank 424 is connected to provide fuel to run a fuel based generator that comprises the excitation source 418. The excitation source 418 is electrically connected to a guided surface waveguide probe 421. The fuel storage tank 424, the excitation source 418, and the guided surface waveguide probe 421 are mounted to a rigid frame 427. The rigid frame 427 allows for secure transportation of the mobile guided surface wave transmit station 400.

In this embodiment, the rigid frame 427 is an intermodal container with a retractable top. In other embodiments, the rigid frame 427 may comprise a railroad car, a truck bed, a trailer, a pallet, or another rigid structure for transportation, as can be appreciated. Alternatively, the rigid frame 427 may comprise a network of metal beams or other structure. The rigid frame 427 may be movable or supportable without damaging the mobile guided surface wave transmit station 400, for example by using a crane, forklift, helicopter, or other machinery. In an additional alternative, wheels may be included to facilitate manual movement of the mobile guided surface wave transmit station 400.

In one embodiment, the guided surface waveguide probe 421 is on a telescoping support structure so that it may be compacted for storage and transport, but may raise its charge terminal T to the extent necessary to transmit a guided surface wave with suitable efficiency. In alternative embodiments, the charge terminal T may be affixed to a boom structure that facilitates lifting the charge terminal T into the air. Alternatively, the charge terminal T may be lifted using a quantity of lifting gas in an aerostat. The telescoping structure, boom structure, or any other structures used to lift the charge terminal T to a desired height as described above are nonconductive structures. Such structures may be constructed, for example, using non-conductive materials such as fiberglass, plastics, and other materials.

Referring specifically to FIG. 24A, shown is an illustration depicting an example of a power reception kit 429. The power reception kit 429 may embody one of the guided surface wave receive stations 406 shown in FIG. 21. According to one embodiment, the components of the power reception kit 429 include an inflatable charge terminal 430 with a connector 431, a compressed gas container 433, a connection wire 436 with connectors 437, a compactable support 439, a ground electrode 442 with a connector 443, power output circuitry 445 with connectors 446, and possibly other components. Each component in the power reception kit 429 is designed to be compatible for use with the other components of the kit. The various connectors may be male, female, hermaphroditic or genderless and may comprise plugs, receptacles, or other connectors to quickly and securely attach and detach the kit components as can be appreciated. Additionally, the connectors may be employed for structural or electronic connections or both.

In certain embodiments, such as for high frequency or low power applications, the components of the power reception kit 429 may fit in a suitcase or a small bag. In other embodiments, such as for low frequency or high power applications, the components of the power reception kit 429 may be transported by a vehicle.

The inflatable charge terminal 430 is designed to be used as part of a guided surface wave receive structure with a tuned resonator and a charge terminal consistent with the embodiments and concepts described herein. This embodiment shows the inflatable charge terminal 430 with a connector 431 to quickly and securely attach to and detach from compatible connectors, for example, at least one of the connectors 437 of the connection wire 436. In various embodiments, the inflatable charge terminal 430 may be made of a conductive material such as aluminum foil, or may be coated with a conductive material such as an aluminum coating, such that the inflatable charge terminal 430 has a self-capacitance as discussed above. When inflated, the inflatable charge terminal 430 may be substantially spherical in shape. Other embodiments may be any shape that allows for a self-capacitance in the inflatable charge terminal 430. Other embodiments may be not be inflatable. As such, a charge terminal in a power reception kit may comprise a conductive sphere, a disc, a loop, or other form of charge terminal as can be appreciated. In various embodiments, the charge terminal 430 may be increased in size, for example, to decrease the likelihood of electrical discharge as discussed above. Note that an inflatable charge terminal may provide for greater compactness when transported as part of a disassembled power reception kit 429.

The compressed gas container 433 includes compressed gas that is used, for example, to inflate the inflatable charge terminal 430. The gas may be any type of compressed gas including compressed air or other gasses as can be appreciated. Alternatively, the compressed gas container 433 may include a lifting gas such as hydrogen, helium, or other lifting gas.

The connection wire 436 is used to connect the inflatable charge terminal 430 to the power output circuitry 445. Of the various embodiments, the connection wire 436 may be of any conductive material and of any gauge or wire size capable of carrying the required power, and may or may not have a protective jacketing, as can be appreciated. In some embodiments, the connection wire 436 may have connectors 437 to quickly and securely attach to and detach from, for example, the compatible connector 431 of the inflatable charge terminal 430 and at least one of the compatible connectors 446 of the power output circuitry 445.

The compactable support 439 in the power reception kit 429 is designed to support the inflatable charge terminal 430 above a lossy conductive medium, such as the earth. In one embodiment, the compactable support 439 may be embodied as a series of rods that are connected end-to-end. A bottom of the compactable support 439 can be secured in the ground, and a top of the compactable support 439 can be connected to the inflatable charge terminal 430 to support it. In other embodiments, the top of the compactable support 439 may be further supported using guy wires as can be appreciated. In some embodiments, the compactable support 439 may additionally be used as a ground electrode. The bottom of the compactable support 439 may additionally be secured in a base such as a tripod or other base as can be appreciated. The compactable support 439 may additionally be an extendable or telescoping rod, extendable manually, hydraulically, pneumatically, or otherwise as can be appreciated. In some embodiments, the compactable support have connectors to quickly and securely attach to and detach from, for example, the a compatible connector on the inflatable charge terminal 430, or a compatible connector on the power output circuitry 445. These connectors may comprise structural connectors and/or electrical connections, for example, an electrical connector to connect to the power output circuitry in order to use the compactable support 439 as a ground electrode.

The ground electrode 442 has the connector 443 that is designed to be connectible to at least one of the connectors 446 of the power output circuitry 445. The ground electrode 442 may be formed of any conductive material of any shape, and may be one of a plurality of ground electrodes as can be appreciated.

The power output circuitry 445 has connectors 446 to quickly and securely attach to and detach from the connector 437 of the connection wire 436 and the connector 443 of the ground electrode 442. In further embodiments, additional connectors may be employed as discussed for structural or electronic connections. In this embodiment, the power output circuitry 445 is used as part of a guided surface wave receive structure with a tuned resonator and a charge terminal as discussed above and also shown in FIG. 18B. Thus, the power output circuitry 445 is shown in FIG. 24A with a plurality of taps on a coil in the output circuitry. As discussed above, to tune reception of a guided surface wave, the phase delay should be equal to the wave tilt for the purpose of mode-matching to the charge terminal at the frequency of operation. The phase can be adjusted by varying the tap position on the coil, and/or by including a plurality of predefined taps along the coil and switching between the different predefined tap locations to maximize efficiency.

Additionally, the power output circuitry 445 is designed to connect to and power an electrical load, and may comprise AC/DC converters, AC/AC converters and other circuitry in order to power a variety of electrical loads as can be appreciated. Further, the power output circuitry 445 may have regulators that limit the power that can be delivered by the power reception kit 429.

FIG. 24B is an illustration of an example of the power reception kit 429 in FIG. 24A after being fully assembled and connected to power a load 415. The individual components are as described above for FIG. 24A.

Next, referring to FIG. 25A, shown is an illustration depicting a power reception kit 450 according to the various embodiments of the present disclosure. In this embodiment, the components of the power reception kit 450 are as follows: an inflatable aerostat 451, a compressed lifting gas container 454, a linear probe wire 455, a ground electrode 456, and power output circuitry 445. Each component in the power reception kit 450 is designed to be compatible for use with the other components of the power reception kit 450. The various components of the power reception kit 450 may also have connectors to quickly and securely attach and detach from other components. The various connectors may be male, female, hermaphroditic or genderless and may comprise plugs, receptacles, or other connectors to quickly and securely attach and detach the kit components as can be appreciated. Additionally, the connectors may be employed for structural or electronic connections or both.

The inflatable aerostat 451 is designed for connection to the linear probe wire 455, and to lift the weight of the linear probe wire 455 in order to position the linear probe wire 455 vertically to receive electrical energy from a guided surface wave. In some embodiments, the inflatable aerostat 451 may have connectors to quickly and securely attach and detach from compatible connectors of the linear probe wire 455.

The compressed lifting gas container 454 will contain a lifting gas suitable for inflating the aerostat 451. The lifting gas should be a gas suitable for lifting such as hydrogen, helium, ammonia, or another lifting gas as can be appreciated.

The linear probe wire 455 is designed for connection to the inflatable aerostat 451 and the power output circuitry 445 and for use as a linear probe in a guided surface wave receive structure (e.g., FIG. 18A) as discussed above. To that end, the linear probe wire 455 may have connectors as discussed above, for example, to structurally attach the linear probe wire 455 to compatible connectors of the inflatable aerostat 451 and to structurally and electronically attach to compatible connectors of the power output circuitry 445. In such an arrangement, the length of the linear probe wire 455 may be optimized. For example, for a maximum voltage with a minimum of wire, length of the linear probe wire 455 should be about one quarter of the wavelength of the guided surface wave. A relatively high impedance as seen from the ground will set the voltage maximum such that it is at the top of the linear probe wire 455, and a relatively low impedance as seen from the ground will set the voltage maximum such that it is at the bottom of the linear probe wire 455.

In some embodiments, the linear probe wire 455 may be thickest at the bottom and thinnest at the top, such that the voltage is highest at the top and the current is lowest at the bottom. In such an arrangement, the weight that the inflatable aerostat 451 must lift is minimized while the current output at the bottom of the linear probe wire 455 is maximized. In further embodiments, the linear probe wire 455 may be a segmented linear probe wire employing a thin wire segment at the top, connected to successively thicker wire segments towards the bottom of the segmented linear probe wire.

The ground electrode 456 is designed for connection with the power output circuitry 445 and may be of any conductive material of any shape, and may be one of a plurality of ground electrodes as can be appreciated. To that end, the ground electrode 456 may have connectors as discussed above to structurally and electronically attach the ground electrode 456 to compatible connectors of the power output circuitry 445.

The power output circuitry 445 is used as part of a guided surface wave receive structure with a linear probe. In order to maximize power output, the load 415 should be substantially impedance matched to the linear probe, here, the conducting wire 455.

Additionally, the power output circuitry 445 is designed to connect to the load 415, and may comprise AC/DC converters, AC/AC converters and other circuitry in order to properly feed a variety of loads as can be appreciated. Further, the power output circuitry 445 may have regulators that limit the power that can be delivered by the system. Also, of the various embodiments, the power output circuitry 445 may include a controller that can be used to control the efficiency of reception or enable or otherwise control the power reception kit 450.

FIG. 25B is an illustration of an example of the power reception kit 450 in FIG. 25A, fully assembled and connected to power the load 415. The individual components are as described above in FIG. 25A.

Next, referring to FIG. 26A, shown is an illustration depicting a power reception kit 458 according to the various embodiments of the present disclosure. In this embodiment, the components of the power reception kit 458 are as follows: a telescoping antenna 459, a ground electrode 460, and power output circuitry 445. Each component in the power reception kit 458 is designed to be compatible for use with the other components of the kit. The various components of the power reception kit 458 may also have connectors to quickly and securely attach and detach from other components. The various connectors may be male, female, hermaphroditic or genderless and may comprise plugs, receptacles, or other connectors to quickly and securely attach and detach the kit components as can be appreciated. Additionally, the connectors may be employed for structural or electronic connections or both.

The telescoping antenna 459 is designed for connection to the power output circuitry 445 and for use as a linear probe guided surface wave receive structure (e.g., FIG. 18A) as discussed above. To that end, the telescoping antenna 459 may have connectors as discussed above, for example, to structurally and/or electronically attach the telescoping antenna 459 to compatible connectors of the power output circuitry 445. In this arrangement, the length of the telescoping antenna 459 may be optimized. For example, for a maximum voltage with the smallest antenna, length of the telescoping antenna 459 may be one quarter of the wavelength of the guided surface wave. A relatively high impedance as seen from the ground will set the voltage maximum to the top of the telescoping antenna 459, and a relatively low impedance as seen from the ground will set the voltage maximum such that it is at the bottom of the telescoping antenna 459. The power output circuitry 445 in this embodiment is substantially similar to the power output circuitry 445 described above with reference to FIG. 25A.

FIG. 26B is an illustration of an example of the power reception kit 458 in FIG. 26A, fully assembled and connected to power a load 415. The individual components are as described above for FIG. 26A.

Next, referring to FIG. 27A, shown is an illustration depicting a power reception kit 462 according to the various embodiments of the present disclosure. In this embodiment, the components of the power reception kit 462 are as follows: a magnetic coil 463 and power output circuitry 445. Each component in the power reception kit 462 is designed to be compatible for use with the other components of the kit.

The magnetic coil 463 is designed for use as a guided surface wave receive structure (e.g., FIG. 19) as discussed above. In order to facilitate reception and/or extraction of electrical power from a guided surface wave, the magnetic coil 463 may be positioned so that the magnetic flux of the guided surface wave passes through the magnetic coil 463, thereby inducing a current in the magnetic coil 463. For example, the magnetic coil 463 may be positioned manually or adjusted using a controller such as one in the power output circuitry 445. The magnetic coil 463 may also have connectors as discussed above to quickly and securely attach and detach from compatible connectors on the power output circuitry 445.

The power output circuitry 445 is designed to connect to the electrical load 415, and may comprise AC/DC converters, AC/AC converters and other circuitry in order to properly feed a variety of loads as can be appreciated. The power output circuitry 445 may maximize power output to a load by substantially impedance matching the electrical load and the magnetic coil 463. Further, the power output circuitry 445 may have regulators that limit the power that can be delivered by the system. Also, of the various embodiments, the power output circuitry 445 may include a controller that can be used to control the efficiency of reception or enable, or otherwise control, the power output.

FIG. 27B is an illustration of an example of the power reception kit 462 in FIG. 27A, fully assembled and connected to power the load 415. The individual components are as described above in FIG. 27A.

Referring now to FIG. 28A, shown is an illustration of a plurality of interconnected power reception kits 465A-D. Each of the plurality of interconnected power reception kits 465A-D is receiving electrical power from its own a guided surface wave receive structure R.

In this embodiment, an AC output port of a power reception kit 465A is coupled to a load 415, which is an AC load. A DC interconnection port of the power reception kit 465A is coupled to a DC output port of a power reception kit 465B. A DC interconnection port of the power reception kit 465B is coupled to a DC output port of a power reception kit 465C. A DC interconnection port of the power reception kit 406C is coupled to a DC output port of a power reception kit 465D, and so on. Any number of power reception kits are connectable in this way.

In this illustration, each of the plurality of interconnected reception kits 465A-D feeds its available power to the power reception kit shown below it. In this embodiment, the power reception kits 465B, 465C, and 465D are only using DC inputs and outputs. Their interconnection may electrically be as simple as a large bus, provided they are all operating at substantially the same voltages. In some embodiments, other isolations and protections such as regulators and diodes may be used. In further embodiments, the interconnection of the plurality of reception kits 465A-D may be via an AC interconnection port that matches one or more of voltage, frequency, or phase. The power reception kit 465A is connected to the power reception kits 465B, 465C, and 465D via its DC interconnection port. That DC power is converted to AC with an inverter in the power reception kit 465A that is large enough to power the load 415, as well as accept the power from the plurality of power reception kits 465A-D.

For example, if each power reception kit is rated to receive 1 KVA via its guided surface wave receive structure, in this configuration the power reception kit 465A should be rated to accept at least 3 kVA through its interconnection port and rated to output at least 4 kVA. In some embodiments the power reception kits 465A-D may all be substantially similar and have substantially similar power ratings. In other embodiments, the power reception kit 465A may have the highest power ratings, with each reception kit 465B, 465C, and 465D having successively lower power ratings.

Referring now to FIG. 28B, shown is an example where an interconnected plurality of power reception kits 466A and an interconnected plurality of power reception kits 466B pass power inversion over to an external converter/inverter 467. The external converter/inverter 467 may convert AC to DC, AC to AC, or DC to AC, of the various embodiments. In one embodiment, the converter/inverter 467 is inverting the DC received from the plurality of interconnected power reception kits 465A-D to AC in order to feed a load 415 that is an AC load. This configuration may be used, for example, to power large loads. For example, if each power reception kit in the plurality of power reception kits 466A-B is rated to receive 1 kVA through its guided surface wave receive structure, accept 2 kVA at its interconnection port, and output 3 KVA, then the converter/inverter 467 should be rated to accept at least 6 kVA through an interconnection port and output at least 6 kVA to a load 415. The converter/inverter 467 may be considered an additional component of a power reception kit.

Referring next to FIG. 29, shown is a diagram that illustrates an example of power output circuitry 445 of a power reception kit such as those of FIGS. 25 through 27, according to the various embodiments of the present disclosure.

Here, a DC input port 469 is connected to an input of a DC paralleling circuit 472. A DC output port 475 is connected to: an output of the DC paralleling circuit 472, an input of a DC/AC inverter 478, and an output of an AC/DC converter 481. Thus, the output of the AC/DC converter 481 is connected to the DC paralleling circuit 472, the DC output port 475, and the DC/AC inverter 478. An output of the DC/AC inverter 478 is connected to a primary of a transformer 484. A secondary of the transformer 484 is connected to an AC output port 487. An AC output port 490 is connected to a secondary of a transformer 493. An AC input port 496 is connected to an input of an AC paralleling circuit 499. An output of a power regulator 503 is connected to an input of the AC/DC converter 481, a primary of the transformer 493, and an output of the AC paralleling circuit 499. An input of the power regulator 503 is connected to an output of an impedance matching circuit 506. An input of the impedance matching circuit 506 is connected to a switch 512. The switch 512 is connected to input terminals 515, and may open and close the circuit pathway between the impedance matching circuit 506 and the input terminals 515. In this embodiment, the input terminals 515 are for connection to output terminals of a guided surface wave receiver R. The input terminals 515 may comprise connectors as discussed above to quickly and securely attach and detach from other components of a power reception kit.

The power output circuitry 445 has a computing device 516, communicatively connected to the regulator 503, the impedance matching circuit 506, and the switch 512. In other embodiments, the computing device 516 may additionally be communicatively connected to components such as, for example, converters, inverters, transformers, interconnecting circuits, paralleling circuits, or other components of the power output circuitry 445.

The computing device 516 may be representative of multiple controllers as can be appreciated. Here, of the various embodiments, the computing device 516 comprises a processor 518, a memory 521, a GPS receiver 524, and an I/O interface 527, all of which are coupled to a local bus 530 which may include a data bus with an accompanying control/address bus as can be appreciated by those with ordinary skill in the art. In this respect, the computing device 516 may comprise a processor circuit as is typical for computer systems as can be appreciated. The processor 518 may be representative of multiple processors, and the memory 521 may be representative of multiple memories. A control system 533 is shown stored in the memory 521 to be executed by the processor 518 to adjust the operation of the computing device 516. The I/O interface 527 is shown connected to an antenna for communications and an input device 536. The input device 536 may be, for example, a keyboard, mouse or other input device. According to the various embodiments, the I/O interface 527 may also include multiple network interfaces for communications through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network.

A battery 540 is shown in the computing device 516 to power the computing device 516 and other components of the power output circuitry 445. The battery 540 allows for preliminary power, setup, and control of the power output circuitry before any power is received, for example, from a guided surface wave receive structure. The power output circuitry 445 may also be configured to charge the battery 540 once power is received.

The power output circuitry 445 is configured to take an input of power from a guided surface wave receive structure at the power input terminals 515 and effectively output that power to an electrical load through one of its output ports. Here, the DC output port 475, the AC output port 487, or the AC output port 490. In order to maximize power to the electrical load, the power output circuitry 445 has an impedance matching circuit 506 configured to effect a conjugate-match condition between the guided surface wave receive structure and the electrical load. The impedance matching circuit is controllable by the computing device 516 through the I/O interface 527.

In order to control maximum power output, the power output circuitry 445 has a regulator 503 configured to control the maximum voltage and current allowed to flow to the electrical load. The regulator 503 may also act to set a steady output current and or voltage.

In most cases, guided surface wave receive structures will output AC power. In order to deliver this AC power to a DC load, the power output circuitry 445 has an AC/DC converter 481. The AC/DC converter 481 may additionally comprise a DC voltage regulator as can be appreciated. Also, the AC/DC converter 481 may convert power received from the AC paralleling circuit 499 to DC power as will be discussed below.

The AC power from a guided surface wave receive structure has a frequency relative to the frequency of the received guided surface wave. This is unlikely to be an operational frequency of an AC load. The power output circuitry 445 has a DC/AC inverter 478 that may convert the DC power from the AC/DC converter 481 into AC power at a common operational frequency such as 50 Hz or 60 Hz. The DC/AC inverter 478 may additionally convert power received from the DC paralleling circuit 472 to AC power as will be discussed below.

The DC paralleling circuit 472 allows interconnection of a plurality of power reception kits 465A-D, for example, as shown in FIG. 28. For example, the DC paralleling circuit 472 may be employed to ensure that the power received through the DC input port 469 is the same DC voltage as the output of the AC/DC converter 481. The DC paralleling circuit 472 and the power output circuitry 445 may additionally have diodes or other components to prevent current backflow. The DC paralleling circuit 472 is an interconnection circuit. In other embodiments, the interconnection circuit may instead be a series circuit or other configuration for interconnection as can be appreciated. In other embodiments, the DC paralleling circuit 472 may be communicatively connected to the computing device 516 to monitor and configure input and output voltages of one or more interconnected power reception kits in communication. If the power reception kits 465A-D, for example, are feeding a single load, the power output circuitry 445 may also be configured to lock all outputs other than those used for interconnection in order to minimize unnecessary power use.

The AC paralleling circuit 499 allows interconnection of the plurality of power reception kits 465 as seen in FIG. 28. For example, the AC paralleling circuit 499 may be employed to ensure that the power received through the AC input port 496 has the same AC voltage, frequency, and phase as the output of the regulator 503. Recall that this frequency would be relative to the frequency of the received guided surface wave.

In the configuration of this embodiment, the AC input port 496 is designed to take input from other power reception kits that are configured to a similar frequency. The AC paralleling circuit 499 is a power source interconnection circuit. In other embodiments, the interconnection circuit may instead be a series circuit or other configuration for interconnection of power sources as can be appreciated. In other embodiments, the AC paralleling circuit 499 may be communicatively connected to the computing device 516 to monitor and configure input and output voltage, phase, or frequency of one or more interconnected power reception kits in communication.

In other embodiments, the power output circuitry 468 may be configured for a specific purpose requiring a greater or lesser number of output ports. For example, the power output circuitry 445 may be configured to output more than one DC voltage and more than one AC frequency at more than one AC voltage. Alternatively, the power output circuitry 445 may be configured to output only DC voltage, allowing for the elimination of components such as the DC/AC inverter 484. Further, the power output circuitry 445 may allow for interconnection only via the AC paralleling circuit 499, or only via the DC paralleling circuit 472.

In this way, the power output circuitry 445 allows for receiving electrical power from guided surface wave receive structures, effectively outputting this power to an electrical load, and interconnection of a plurality of power output circuitries to an electrical load. The computing device 516 may allow for coordination via the communications networks discussed above, between the plurality of power output circuitries. The computing device 516 may also be in communication with a central system that may, for example, enable authorized power reception kits or refuse to enable unauthorized power reception kits, or limit the power output of the power reception kits.

Referring now to FIG. 30, shown is a flow chart that provides one example of the operation of the control system 533 of a power reception kit according to an embodiment of the present invention. Alternatively, the flow chart in FIG. 30 may be viewed as depicting steps of an example of a method implemented in a server, computer system, or like device. The functionality of the control system 533 depicted by the example flow chart of FIG. 30 may be implemented, for example, in an object-oriented design or in some other programming structure.

Assuming the functionality is implemented in an object-oriented design, then each block represents functionality that may be implemented in one or more methods that are encapsulated in one or more objects. The control system 533 may be implemented using any one of a number of programming languages such as, for example, C, C++, or other programming languages.

Beginning with box 541, the control system 533 will send status information to a central system. The status information may include, for example, GPS location, user information, identification information, power usage information, load requirements, or other information. In other embodiments, there may be no central system.

Next, in box 543, the control system 533 will check whether the power reception kit has authorization to operate. For example, the authorization may be granted by password, GPS location, biometrics, or otherwise. The authorization may be checked using only the computing device 516, or may require communication with a central system using any of the communications networks discussed above. If the power reception kit has authorization to operate then the control system 533 will continue to box 546. If not, then the control system 533 will proceed to box 549.

In box 546, the control system 533 enables the power reception kit and continues to box 552.

In box 549, the control system 533 shuts down the power reception kit and proceeds back to box 541. For example, the control system 533 may open the switch 512 in order to shut it down. In other embodiments, the shutdown may only limit the power output of the power reception kit, or may use the status information to determine whether to shut down or limit power output of the power reception kit.

In box 552, the control system 533 checks to see whether the authorization has timed out. For example, the power reception kit may be authorized for continuous use and have no time out, or the power reception kit may be authorized only for a certain amount of time. If authorization has timed out, then the control system 533 will proceed to box 549. If not, then the control system will continue to box 555.

In box 555, the control system 533 checks to see whether a receiver shutdown command has been issued or otherwise scheduled. For example, the central system may need to shut down some receivers in order to free up available power elsewhere. In this situation, the shutdown command will be sent from the central system. In another example, all power generation may be on a predetermined schedule and the shut down time stored in the memory 521 of the computing device 516. In yet another example the central system may desire to revoke authorization for any other reason. If a shutdown command has been issued or otherwise scheduled, then the control system 533 will proceed to box 549 and if not, then the control system 533 will continue to box 558.

In box 558, the control system 533 will check to see whether the power reception kit is overloaded. For example, the computing device 516 may monitor power output using the regulator 503 to determine whether the reception kit is overloaded. Alternatively, the computing device 516 may send load information to the central system, that may determine and communicate to the computing device 516 that an overload condition exists. If the power reception kit is overloaded, then the control system 533 will proceed to box 549 and if not, then the control system 533 will continue to box 561.

In box 561, the control system 533 will check to see whether a lock outputs command has been issued or otherwise scheduled. For example, if the power reception kit is interconnected with other power reception kits to power a large load or an important load, it may be desirable to lock certain output ports to prevent unnecessary power usage, or less important power usage. In another example, such as when battery power is low, it may be desirable to lock all power output ports rather than completely disable the power reception kit. If a lock outputs command has been issued or otherwise scheduled, then the control system 533 will proceed to box 564 and if not, then the control system 533 will continue to box 541.

In box 564, the control system 533 locks outputs of the power reception kit and then continues to box 541.

With reference to FIG. 29, shown is a schematic block diagram of the computing device 516 according to an embodiment of the present disclosure. Each computing device 516 includes at least one processor 518 and memory 521, both of which are coupled to a bus 530. To this end, the computing device 516 may comprise, for example, at least one server, computer or like device. The bus 530 may comprise, for example, a data bus with an accompanying address/control bus or other bus structure as can be appreciated.

Stored in the memory 521 are both data and several components that are executable by the processor 518. In addition, an operating system may be stored in the memory 521 and executable by the processor 518. In particular, stored in the memory 521 and executable by the processor 518 are, for example, the control system 533 and potentially other applications.

It is understood that there may be other applications that are stored in the memory 521 and are executable by the processor 518 as can be appreciated. Where any component discussed herein is implemented in the form of software, any one of a number of programming languages may be employed such as, for example, C, C++, C≠, Objective C, Java®, JavaScript®, Perl, PHP, Visual Basic®, Python®, Ruby, Flash®, or other programming languages.

A number of software components are stored in the memory 521 and are executable by the processor 518. In this respect, the term “executable” means a program file that is in a form that can ultimately be run by the processor 518. Examples of executable programs may be, for example, a compiled program that can be translated into machine code in a format that can be loaded into a random access portion of the memory 521 and run by the processor 518, source code that may be expressed in proper format such as object code that is capable of being loaded into a random access portion of the memory 521 and executed by the processor 518, or source code that may be interpreted by another executable program to generate instructions in a random access portion of the memory 521 to be executed by the processor 518, etc. An executable program may be stored in any portion or component of the memory 521 including, for example, random access memory (RAM), read-only memory (ROM), hard drive, solid-state drive, USB flash drive, memory card, optical disc such as compact disc (CD) or digital versatile disc (DVD), floppy disk, magnetic tape, or other memory components.

The memory 521 is defined herein as including both volatile and nonvolatile memory and data storage components. Volatile components are those that do not retain data values upon loss of power. Nonvolatile components are those that retain data upon a loss of power. Thus, the memory 521 may comprise, for example, random access memory (RAM), read-only memory (ROM), hard disk drives, solid-state drives, USB flash drives, memory cards accessed via a memory card reader, floppy disks accessed via an associated floppy disk drive, optical discs accessed via an optical disc drive, magnetic tapes accessed via an appropriate tape drive, and/or other memory components, or a combination of any two or more of these memory components. In addition, the RAM may comprise, for example, static random access memory (SRAM), dynamic random access memory (DRAM), or magnetic random access memory (MRAM) and other such devices. The ROM may comprise, for example, a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or other like memory device.

Also, the processor 518, may represent multiple processors 518 and/or multiple processor cores and the memory 521 may represent multiple memories 521 that operate in parallel processing circuits, respectively. In such a case, the bus 530 may be an appropriate network that facilitates communication between any two of the multiple processors 518, between any processor 518 and any of the memories 521, or between any two of the memories 521, etc. The bus 530 may comprise additional systems designed to coordinate this communication, including, for example, performing load balancing. The processor 518 may be of electrical or of some other available construction.

Although the control system 533 and other various systems described herein may be embodied in software or code executed by general purpose hardware as discussed above, as an alternative the same may also be embodied in dedicated hardware or a combination of software/general purpose hardware and dedicated hardware. If embodied in dedicated hardware, each can be implemented as a circuit or state machine that employs any one of or a combination of a number of technologies. These technologies may include, but are not limited to, discrete logic circuits having logic gates for implementing various logic functions upon an application of one or more data signals, application specific integrated circuits (ASICs) having appropriate logic gates, field-programmable gate arrays (FPGAs), or other components, etc. Such technologies are generally well known by those skilled in the art and, consequently, are not described in detail herein.

The flowchart of FIG. 30 shows the functionality and operation of an implementation of portions of the control system 533. If embodied in software, each block may represent a module, segment, or portion of code that comprises program instructions to implement the specified logical function(s). The program instructions may be embodied in the form of source code that comprises human-readable statements written in a programming language or machine code that comprises numerical instructions recognizable by a suitable execution system such as a processor in a computer system or other system. The machine code may be converted from the source code, etc. If embodied in hardware, each block may represent a circuit or a number of interconnected circuits to implement the specified logical function(s).

Although the flowchart of FIG. 30 shows a specific order of execution, it is understood that the order of execution may differ from that which is depicted. For example, the order of execution of two or more blocks may be scrambled relative to the order shown. Also, two or more blocks shown in succession in FIG. 30 may be executed concurrently or with partial concurrence. Further, in some embodiments, one or more of the blocks shown in FIG. 30 may be skipped or omitted. In addition, any number of counters, state variables, warning semaphores, or messages might be added to the logical flow described herein, for purposes of enhanced utility, accounting, performance measurement, or providing troubleshooting aids, etc. It is understood that all such variations are within the scope of the present disclosure.

Also, any logic or application described herein, including the control system 533, that comprises software or code can be embodied in any non-transitory computer-readable medium for use by or in connection with an instruction execution system such as, for example, a processor in a computer system or other system. In this sense, the logic may comprise, for example, statements including instructions and declarations that can be fetched from the computer-readable medium and executed by the instruction execution system. In the context of the present disclosure, a “computer-readable medium” can be any medium that can contain, store, or maintain the logic or application described herein for use by or in connection with the instruction execution system.

The computer-readable medium can comprise any one of many physical media such as, for example, magnetic, optical, or semiconductor media. More specific examples of a suitable computer-readable medium would include, but are not limited to, magnetic tapes, magnetic floppy diskettes, magnetic hard drives, memory cards, solid-state drives, USB flash drives, or optical discs. Also, the computer-readable medium may be a random access memory (RAM) including, for example, static random access memory (SRAM) and dynamic random access memory (DRAM), or magnetic random access memory (MRAM). In addition, the computer-readable medium may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or other type of memory device.

Further, any logic or application described herein, including the control system 533, may be implemented and structured in a variety of ways. For example, one or more applications described may be implemented as modules or components of a single application. Further, one or more applications described herein may be executed in shared or separate computing devices or a combination thereof. For example, a plurality of the applications described herein may execute in the same computing device, or in multiple computing devices in the same computing environment. Additionally, it is understood that terms such as “application,” “service,” “system,” “engine,” “module,” and so on may be interchangeable and are not intended to be limiting.

Disjunctive language such as the phrase “at least one of X, Y, or Z,” unless specifically stated otherwise, is otherwise understood with the context as used in general to present that an item, term, etc., may be either X, Y, or Z, or any combination thereof (e.g., X, Y, and/or Z). Thus, such disjunctive language is not generally intended to, and should not, imply that certain embodiments require at least one of X, at least one of Y, or at least one of Z to each be present.

It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. In addition, all optional and preferred features and modifications of the described embodiments and dependent claims are usable in all aspects of the disclosure taught herein. Furthermore, the individual features of the dependent claims, as well as all optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another. 

Therefore, the following is claimed:
 1. An apparatus, comprising: a rigid frame; and a guided surface waveguide probe coupled to the rigid frame, the guided surface waveguide probe comprising a charge terminal and being configured to launch a Zenneck surface wave along a surface of a lossy conducting medium by generating at least one resultant field that synthesizes a wave front incident with the surface at a complex Brewster angle of incidence of the lossy conducting medium.
 2. The apparatus of claim 1, further comprising an excitation source electrically connected to the guided surface waveguide probe, the excitation source being configured to excite the charge terminal of the guided surface waveguide prove.
 3. The apparatus of claim 2, wherein the excitation source comprises a guided surface wave receive structure, the guided surface wave receive structure being configured to receive electrical energy from a second Zenneck surface wave generated by a remotely-located guided surface wave, and the charge terminal of the guided surface waveguide probe being excited to generate the at least one resultant field via the electrical energy received via the second Zenneck surface wave.
 4. The apparatus of claim 1, further comprising a feed network electrically coupled to the charge terminal, the feed network providing a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with the complex Brewster angle of incidence (θ_(i,B)) associated with the lossy conducting medium in the vicinity of the guided surface waveguide probe.
 5. The apparatus of claim 1, wherein the rigid frame comprises at least one of: an intermodal container, a railroad car, a truck bed, a trailer, a pallet, or a network of beams.
 6. The apparatus of claim 1, further comprising a non-conductive structure coupled to the guided surface waveguide probe, the non-conductive structure being configured to adjust a height of the charge terminal over the lossy conducting medium.
 7. The apparatus of claim 6, wherein the non-conductive structure comprises a telescoping structure or a boom structure.
 8. The apparatus of claim 1, wherein the charge terminal is one of a plurality of charge terminals.
 9. The apparatus of claim 1, wherein a field strength of the guided surface wave decays exponentially as a function of a distance from the guided surface waveguide probe.
 10. A method, comprising: elevating a charge terminal of a guided surface waveguide probe over a lossy conducting medium, the guided surface waveguide probe being mounted on a rigid frame, and the charge terminal being coupled to a non-conductive moveable support structure; and launching a Zenneck surface wave via the guided surface waveguide probe by exciting the charge terminal to generate at least one resultant field that synthesizes a wave front incident with the surface at a complex Brewster angle of incidence of the lossy conducting medium
 11. The method of claim 10, wherein the Zenneck surface wave comprises a first Zenneck surface wave and the guided surface waveguide probe comprises a locally-located guided surface waveguide probe, and the method further comprises receiving, via a guided surface wave receive structure mounted on the rigid frame, a second Zenneck surface wave generated by a remotely-located guided surface wave.
 12. The method of claim 11, wherein electrical energy of the second Zenneck surface wave is used to excite the charge terminal.
 13. The method of claim 10, further comprising providing, to the charge terminal, a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with the complex Brewster angle of incidence (θ_(i,B)) associated with the lossy conducting medium in the vicinity of the guided surface waveguide probe.
 14. The method of claim 10, wherein the non-conductive moveable support structure comprises a telescoping structure, and elevating the charge terminal comprises extending the telescoping structure.
 15. The method of claim 10, wherein the rigid frame comprises at least one of: an intermodal container, a railroad car, a truck bed, a trailer, a pallet, or a network of beams.
 16. An apparatus, comprising: a rigid frame; an excitation source mounted on the rigid frame; and a guided surface waveguide probe electrically coupled to the excitation source and mounted to the rigid frame, the guided surface waveguide probe being configured to launch a Zenneck surface wave along a surface of a lossy conducting medium by generating at least one resultant field that synthesizes a wave front incident with the surface at a complex Brewster angle of incidence of the lossy conducting medium.
 17. The apparatus of claim 16, further comprising a matching network mounted to the rigid frame, the matching network coupling the excitation source to the guided surface waveguide probe, and the matching network being configured to effect a conjugate match for maximum power transfer.
 18. The apparatus of claim 16, wherein the guided surface waveguide probe comprises a charge terminal coupled to a moveable non-conductive structure configured to adjust a height of the charge terminal over the lossy conducting medium.
 19. The apparatus of claim 16, wherein the rigid frame comprises at least one of: an intermodal container, a railroad car, a truck bed, a trailer, a pallet, or a network of beams.
 20. The apparatus of claim 16, wherein the Zenneck surface wave comprises a first Zenneck surface wave, the excitation source comprises a guided surface wave receive structure, the guided surface wave receive structure is configured to receive electrical energy from a second Zenneck surface wave generated by a remotely-located guided surface wave, and the charge terminal of the guided surface waveguide probe is excited to generate the at least one resultant field via the electrical energy received via the second Zenneck surface wave. 